This truck driver gear shift is so smooth I’m not even sure I’d count it as the truck driver gear shift, but I guess technically it is:

Roxette – Listen to Your Heart

It’s in Bm, but let’s say it’s in Am to simplify notation. The first two verses and choruses, and the solo, are all in Am. Then there’s a brief bridge in D, and I would definitely say that’s a real key change, before it goes back to the chorus, this time in E. The solo → bridge → chorus transition looks something like:

CHORUS (in Am)
Listen to your h[Am]eart[F] when he's c[C]alling for y[G]ou.
Listen to your h[Am]eart[F] there's nothing e[C]lse you can d[G]o.
I don't kn[C]ow where you're g[G]oing and [F]I don't know [C]why,
but listen to your h[Am]eart[F] before[G] you tell him goodbye[Am].

SOLO (in Am)
[Am] [F] [G] [Am] x 2
[C] [G] [F] [C]
[Am] [F] [G] [G]

BRIDGE (in D) (note the G → D cadence to smoothly move into the bridge)
[D] And there are voices that want to be heard.
[Bm] So much to mention but you can't find the words.
[A] The scent of magic, the be[G]auty that's been
[A] when love was wilder [B] than the wind.

CHORUS (in E)
Listen to your h[C#m]eart[A] when he's c[E]alling for y[B]ou
Listen to your h[C#m]eart,[A] there's nothing [E]else you can [B]do
I don't kn[E]ow where you're g[B]oing and [A]I don't know w[E]hy,
but listen to your h[C#m]eart[A] bef[B]ore you tell him goodb[C#m]ye.

I like this because it’s not fancy at all, but still contains two key changes and everything fits together perfectly. It’s good clean Scandinavian pop. Nothing superfluous. Like Ikea furniture.

What other songs do a double key change like this? There must be plenty. Doesn’t have to be the same delta twice, but the second one should obviously not go back to the original key. And it would be better if they weren’t exactly the same.

Closest thing I can think of right now is Total Eclipse of the Heart, but I’m not 100% sure I would allocate two key changes from the start of the verse to the start of the (pre)chorus.

Bonnie Tyler – Total Eclipse of the Heart

Writing in Bm, but it’s really around Bbm.

VERSE (in Bm/D)
[Bm](Turn around...) Every now and then I get a 
[A]little bit lonely and you're never coming 'round.
[Bm](Turn around...) Every now and then I get a
[A]little bit tired of listening to the sound of my tears.

VERSE (still in D but threatening with that bVII)
[D](Turn around...) Every now and then I get a 
[C]little bit angry
     and I know I've got to get out and cry.
[D](Turn around...) Every now and then I get a 
[C]little bit terrified
     but then I see the look in your eyes.

VERSE (now possibly in F or Bb)
[F](Turn around, [Bb]bright eyes...) Every now and then I fall apart.
[F](Turn around, [Bb]bright eyes...) Every now and then I fall apar[A]t.

CHORUS (in F#m/A)
And I [F#m]need you now tonight[D], I [E]need you more than [A]ever...

This song is so good that it counteracts the effect of making me feel like I’m at a middle school dance pretending to be super busy speaking about Important Matters with my Dungeons and Dragons friends while other people are dancing with members of the opposite sex.

But on the double-key-change topic… I can think of a few flavors of double-key-change. As per your definition, we’re discounting songs that just change back to the original key, including songs where the chorus is just in a different key than the verse, which unfortunately takes away my excuse to talk (again) about Alone:

Heart – Alone

…which goes from an Am verse to a Dm chorus at 0:50 (really Bbm to Ebm). Oh, shit, did i just go and talk about it anyway? I guess I’ll have to make it worth your while. File under “chorus and verse in different keys”. Oh, wait, even better, file under “’80s power ballads with the chorus and verse in different keys”. We’ll come back to this.

Your definition would also discount songs where the solo is in a different key than the rest of the song, which was really popular in the ’80s. Usually it was a whole step up. See:

• Megadeth – Foreclosure of a Dream
  (from Gm to Am, just for the solo, at 2:20)
• Guns n’ Roses – Nighttrain
  (from Abm to Bbm, just for the solo, at 1:55)

Now, what if I could deliver for you an ’80s power ballad with the chorus and verse in different keys *and* a solo in yet another key, with a prechorus that I conjecture is in a *fourth* key, all without any awkward prog-y jarring key changes? This would, I think, fit your requirements. I think Ian knows where I’m going now. (That seemingly unnecessary last sentence was itself a pun to which I’ll return.)

And what if I could deliver all that with a video so absurdly ’80s that even if you don’t like the song at all, you’ll sit down with a nice cup of Tang and won’t look away until the video ends on an “’80s face”, which you may not have thought was a thing, but now you know.

I bring you “High Enough” by Damn Yankees:

Damn Yankees – High Enough

After the introductory mess that has nothing to do with the song and appears to be some sort of Dukes of Hazard parody, the song starts in A at 0:25.

Damn Yankees – High Enough (first verse, in A)

[Asus2]I don't wanna hear about it anymore.
It's a [F#m7]shame I've got to live without you	anymore.
There's [Esus4]a fire in my heart,
A [E]pounding in my brain, drivin' me [Dsus2]crazy.

I conjecture that the prechorus is long enough to qualify as a transition to Am, but if you disagree, it doesn’t disqualify the other three keys that occur elsewhere in the song:

Damn Yankees – High Enough (pre-chorus, in Am)

[F] Dont' say good[G]night.
[F] Say you're gonna [Am]stay forever.
[F] Oh [G]woah all the [A]way

The chorus comes in smoothly in D:

Damn Yankees – High Enough (first chorus, in D)

[D]Can you take [A]me hi[Bm]gh enough?
[Em] To fly me over, (fly me [G]over) yester[A]day
[D] Can you take [A]me hi[Bm]gh enough?
[Em] It's never over
And [G]yesterday's just a memory (yesterday's just a memory)

…and back to A for the second verse.

Interestingly something about the bridge *feels* like a key change, but I can’t figure out what it is; it’s clearly in Bm:

Damn Yankees – High Enough (bridge, in Bm)

[Bm]honey (whoa, oh [D] oh oh) I was [A]running for the
[Bm]door (whoa, oh [D] oh oh) the [A]next thing I re-...

Then the solo, as per foreshadowing I gave you above (which I’m diminishing as quality foreshadowing by calling out here), comes in its own key (E) at 3:10:

Damn Yankees – High Enough (solo, in E)

…and back to D for the last chorus. I BEG you to check out the face Ted Nugent makes as the video fades. It’s SO inconsistent with the music, but at the same time so wildly inconsistent with the video theme (which itself is wildly inconsistent with the music) that it could only have been the late ’80s (technically 1990, but supergroups were given permission to extend the ’80s until 1999).

Whew. That’s some multiple key changes. Damn Yankees, IMO, were underrated and sadly short-lived (I own both albums), and might be the most vocal talent ever assembled in a rock band. So to honor their greatness, I’ll give an honorable mention to their other power ballad, “Where You Goin’ Now” (see pun above), which starts in Am:

Damn Yankees – Where You Goin’ Now

…and has a good-old-fashioned gear shift before the last chorus (in D):

Damn Yankees – Where You Goin’ Now (last chorus)

I also could argue that the bridge is in Cm, but it’s close (this progression, bVI → bVII → I (if you think it’s in the major key), or possibly VI → VII → I (if you think it’s in the minor key), straddles the line between the major and its relative minor):

Damn Yankees – Where You Goin’ Now (bridge)

   
[Ab]Now I'm not [Bb]talking about what's [C]good for me
[Ab]And I'm not [Bb]saying how you [C]ought to be
[Ab]But if there is a [Bb]message
[Eb]shining on [Eb/D]through to [Cm]you
[F]Take a little peace of mind [G]and let that love light shine

At the very least, this song gives us lots of:

• Huge ’80s hair
• ’80s guitar faces
• ’80s video girls
• The n-apostrophe-instead-of-the-word-“and” construct, which saw a huge boom in the ’80s
• The use of the “somebody call a doctor” line, which seemed to fade from popularity around 1990
• Singing about being attracted high-school age girls, which had a good and super-creepy 40-year run, but also went out of favor around 1990 (also see Winger: Seventeen, The Beatles: I Saw Her Standing There, Gary Puckett: Young Girl)

On other multiple-key-change fronts, the sequential chromatic key change seems to have fallen out of favor, but way back when…

Mack the knife changes keys with almost every verse. It starts in Bb, up to B at 0:55, C at 1:20, C# at 1:43, D at 2:08, Eb at 2:30. Also that very last chord is possibly non-diatonic, adding one more to the short list of otherwise-non-random songs that end on a non-diatonic chord.

Bobby Darin – Mack the Knife

Similarly, the Peggy Lee recording of “Fever” starts in Bbm, then goes up to Bm at 1:30, then Cm at 2:00.

Peggy Lee – Fever

Oooh, and if “High Enough” was part of a larger class of “chorus and verse in different keys and then something random happens in another key”, maybe the best-known example of this is Layla:

Derek and the Dominos – Layla

This whole song is a bit between keys due to late-60s tape stretch; I’m going to round down. The intro and chorus are in Dm:

Lay[Dm]la,  [Bb]  you [C]got me on my [Dm]knees...

The verse, which has some awesome non-diatonics (actually another bVI → bVII → I, which I can only assume Clapton stole from Damn Yankees and/or Mötley Crüe), is in C#m:

[C#m]What will you do when you get [G#7lonely
[C#m]With nobody [C]waiting [D]by your [E]side [E7]
[F#m]You've been [B]running and [E]hiding much too [A]long,
[F#m]You know it's [B]just your foolish [E]pride.    

And the totally random outro is in C, with a nice bVIIsomething, ergo multiple key changes.

This is sort of cheating, since it’s almost two different songs.

I can’t lie: Ian’s email resulted in me watching a half hour of Roxette videos on my phone, followed by additional non-Roxette YouTube rathole videos. I am now up to date on their bio according to Wikipedia and have seen multiple live versions of The Look. Let’s move on.

Wouldn’t Pinball Wizard also qualify?

I don’t think I can think of anything to top Dan’s Damn Yankees suggestion but I argue that Layla falls into the kind of prog-y territory that should probably be disqualified for this exercise.

P.S. A rathole is like a rabbithole that’s filled with shame and 80’s Videos That Shall Not Be Named.

I think you mean a Ratthole.

Pinball Wizard is more prog-y than I remembered, and makes a good test case for the question of where the line is between “complicated pop song with multiple keys” and “fuck it this is prog so why bother with keys?”.

The intro clearly starts in Bm but resolves nicely to B:

Intro (in Bm)
[Bm][Bmsus][F#7sus][F#7]...

…and the verse I’d say is in B, but one could also argue that it changes key with each line (B, A, G, F#):

First verse (in B)
Ever [Bsus4]since I was a young boy I've [B]played the silver ball.
From [Asus4]Soho down to Brighton, I [A]must've played them all.
But I [Gsus4]ain't seen nothing like him, in [G]any amusement hall.
That [F#sus4]deaf, dumb, and blind kid [F#]sure plays a mean pin[B]ball.

It’s hard to argue that at the third line (over Gsus4 and G) it still feels like there’s a tonal center in B. I don’t know you can say it’s in G for 4 seconds, but hard to say it’s still in B.

Then there’s the instrumental bit one could call a chorus, which I’d say is in E (baa daa dum dum, ba DUM):

Instrumental chorus-like bit (in E)
[B] [A] [D] [E] [B] [A] [D] [E]

Then eventually the chorus is clearly in B:

Chorus (in B)
He's a [E]pin[F#]ball [B]wizard there [E]has to [F#]be a [B]twist...

Then the bridge thing in either D mixolydian or G major:

Bridge (in D mixolydian or G major)
[D]How do you [C]think he [G]does it?
[D]What makes [C]him so [G]good?

Then the last verse definitely starts in D, but like the first verse, it’s hard to argue that it’s still in D by the time we’re at the Bbsus4:

Even [Dsus4]on my favourite table [D]he can beat my best
His dis[Csus4]ciples lead him in, and [C]he just does the rest
He's got [Bbsus4]crazy flipper fingers, [Bb]never seen him fall
That [A7sus4]deaf, dumb, and blind kid [A]sure played a mean pin[D]ball

So the result could be 2 keys (B and Bm/D), 3 keys (B, Bm/D, and E), 8 keys, or “prog”. I vote “prog”.