Formal structure of [154] "The Continuing Story of Bungalow Bill":
          Intro (ind)    0:00-0:08      E minor
          Chorus           0:08-0:24      C major, A major
          Verse 1          0:24-0:46      A minor
          Chorus          0:46-1:01       C major, A major
          Verse 2          1:01-1:24       A minor
          Chorus           1:24-1:40      C major, A major
          Verse 3          1:40-2:01      A minor
          Chorus           2:01-2:16      C major, A major
          Chorus          2:16-2:30       C major, A major
          Chorus          2:30-2:45      C major, A major
          Chorus          2:45-3:00      C major, A major
          Coda (ch)    3:00-3:13        n/a

Comments: Despite the surface-level simplicity of this children's song, the tonal and formal designs are quite sophisticated. 

The introductory flamenco guitar solo is completely independent from the rest of the song (rather similar to the total non sequitur string quartet coda in [144] "Glass Onion").

The three verses are all in A minor; the choruses all alternate from C major to A major. Several Beatles' songs to date have pitted A against C ([48] "Another Girl", [55] "You're Going to Lose That Girl", [122] "Lady Madonna", and [146] "Birthday"), and a couple others have juxtaposed other tonalities that are also a minor third apart ([91] "Here There and Everywhere", and [103] "Lucy in the Sky With Diamonds" both employ G major and B-flat major). Others juxtapose A major with A minor ([35] "Things We Said Today", and [136] "While My Guitar Gently Weeps"), the parallel relationship of which is also used in other songs and other keys ([63] "Norwegian Wood" uses E major and E minor, and [71] "Michelle" uses F major and F minor). The play between A minor, A major, and C major specifically was used in [148] "Happiness is a Warm Gun", and will reappear in the Abbey Road Medley.

Four contiguous choruses ties a record for a Beatles track. And if you include the coda as a chorus (it is an extension of the chorus - but only by the synthesized trombone, played on a mellotron - while every other instrument stops), then it sets a new record. 

 And compared to contiguous verses (which is an extremely common formal design in Beatles music), contiguous choruses are only seldom found - and when they do appear, it's usually at the very end of the song, with the final chorus doubling as the coda: [44b] "Kansas City/Hey Hey Hey Hey" used four contiguous choruses, but they were two different choruses (Chorus #1, Chorus #1 again, Chorus #2, Chorus #2 again); [58] "I'm Down", [73] "Think For Yourself", [90] "Good Day Sunshine", [113] "It's All Too Much", and [114] "All You Need Is Love" all used three contiguous choruses, with the third doubling as the coda; [117] "Blue Jay Way" uses four contiguous choruses, with the fourth doubling as the coda; [133] "Cry Baby Cry" used three contiguous choruses immediately preceding a musically independent coda; and [150] "Savoy Truffle" used two contiguous choruses, with the second doubling as the coda.

[121] "The Inner Light"
The middle 8 is used twice, lasting 8 or 9 measures and 23 or 18 seconds each time (the second iteration transitions to the coda, accounting for the slightly shortened duration), totaling 26.3% (41/156) of the song's duration.

[122] "Lady Madonna"
The middle 8 is used thrice (with the middle of those three replacing vocals with a saxophone solo), lasting 8 measures and 18 or 17 seconds each time, totaling 38.7% (53/137) of the song's duration. The middle 8s also modulate from A Major to C Major.

[123] "Across the Universe"
"Across the Universe" is the 46th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[124] "Hey Bulldog"
"Hey Bulldog" is the 47th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[125] "Revolution 1"
"Revolution 1" is the 48th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[126] "Don't Pass Me By"
"Don't Pass Me By" is the 49th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[127] "Revolution 9"
"Revolution 9" is the 50th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[128] "Blackbird"
The middle 8 is used twice, lasting 4.5 measures and 11.5 seconds each time, totaling 16.7% of the song's duration.

[129] "Everybody's Got Something to Hide Except for Me and My Monkey"
"Everybody's Got Something to Hide Except for Me and My Monkey" is the 51st Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[130] "Good Night"
The middle 8 is used thrice, lasting 4 measures and 14 or 15 seconds each time, totaling 22.3% of the song's duration.

[131] "Ob-La-Di Ob-La-Da"
The middle 8 is used twice, lasting 8 measures and 15 seconds both times, totaling 16.0% of the song's duration. The middle 8 tonicizes IV.

[132] "Revolution"
"Revolution" is the 52nd Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[133] "Cry Baby Cry"
"Cry Baby Cry" is the 53rd Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[134] "Helter Skelter"
"Helter Skelter" is the 54th Beatles original not to use a middle 8. Instead, the chorus serves as contrast to the verses.

[135] "Sexy Sadie"
The middle 8 is used twice, lasting 5.5 measures and averaging 11.5 seconds each time, totaling 11.9% of the song's duration.

[136] "While My Guitar Gently Weeps"
The middle 8 is used twice, lasting 16 measures and 34 seconds each time, totaling 23.69% of the song's duration. The middle 8 modulates from A minor to A major.

[137] "Hey Jude"
The middle 8 is used twice, lasting 11.5 measures and 37.5 seconds each time, totaling 17.5% of the song's duration.

[138] "Mother Nature's Son"
The middle 8 is used twice, lasting 7.5 measures and 21 seconds each time, totaling 25.1% of the song's duration.

[139] "Yer Blues"
The middle 8 is used thrice, lasting 4 measures and averaging about 10 seconds each, totaling 12.1% (29/239) of the song's duration.

[140] "Rocky Raccoon"
The middle 8 is used twice, lasting 8 measures the first time and 9 measures the second (because the second middle 8 concludes the song), averaging 24 seconds each, which totals 22.6% (48/212) of the song's duration.

[141] "Wild Honey Pie"
"Wild Honey Pie" is the 55th Beatles song not to use a middle 8. The novel structure of the song has no need for a middle 8.

[142] "Back in the USSR"
The middle 8 is used twice, lasting 10 measures and 17 seconds each time, totaling 14.7% (24/163) of the song's duration.

[143] "Dear Prudence"
The middle 8 is used once, lasting 5.5 measures and 17 seconds, totaling 7.2% (17/235) of the song's duration.

[144] "Glass Onion"
The middle 8 is used once, lasting 10 measures and 19 seconds, totaling 13.9% (19/137) of the song's duration. This middle 8 concludes with two bars borrowed from the end of verse B.

[145] "I Will"
The middle 8 is used once, lasting 8 measures and 19 seconds, totaling 18.1% (19/105) of the song's duration.

[146] "Birthday"
The middle 8 is used twice, lasting 10 measures and 14 seconds each time, totaling 17.3% (28/162) of the song's duration. The middle 8 modulates form A major to C major.

[147] "Piggies"
The middle 8 is used once, lasting 7 measures and 18 seconds, totaling 14.6% (18/123) of the song's duration.

[148] "Happiness is a Warm Gun"
"Happiness is a Warm Gun" is the 56th Beatles song not to use a middle 8. The novel structure of the song has no need for a middle 8.

[149] "Honey Pie"
The middle 8 is used twice, lasting 8 measures and 13 seconds each time, totaling 16.3% (26/160) of the song's duration.

[150] "Savoy Truffle"
The middle 8 is used twice, lasting 8 measures and averaging 15.5 seconds, totaling 21.1% (31/174) of the song's duration.

[151] "Martha My Dear"
The middle 8 is used twice, with the first iteration incorporating a substantial extension. The first middle 8 (extension included) is 15.5 measures and 42 seconds long; the second is 8.5 measures and 23 seconds long. Combine these two, and the middle 8 totals 43.9% (65/148) of the song's duration.

[152] "Long Long Long"
The middle 8 is used once, lasting 7.5 measures and 26 seconds, totaling 14.1% (26/185) of the song's duration.

[153] "I'm So Tired"
"I'm So Tired" is the 57th Beatles song not to use a middle 8. Instead, the chorus serves as a contrast to the verses.

[154] "The Continuing Story of Bungalow Bill"
"The Continuing Story of Bungalow Bill" is the 58th Beatles song not to use a middle 8. Instead, the chorus serves as a contrast to the verses.

[155] "Why Don't We Do It In The Road"
"Why Don't We Do It In The Road" is the 59th Beatles song not to use a middle 8. (There is no contrast to the verses, with the possible exception of the percussion introduction).

[156] "Julia"
The middle 8 is used once, lasting 5 measures and 18 seconds, totaling 10.2% (18/176) of the song's duration.


Updated cumulative chart:

As a mandolin enthusiast, I was particularly intrigued when I heard what appeared to be a mandolin in the song "The Continuing Story of Bungalow Bill". Upon researching the matter however, I discovered the following:

Though no Beatles recording uses a real mandolin, "The Continuing Story of Bungalow Bill" uses a simulated one. Each verse ("We went out tiger hunting...", "Deep in the jungle...", "The children asked...") is accompanied by a simulated tremoloed mandolin played on the mellotron, an early sample-playback keyboard, a predecessor to the modern synthesizers. This was added by Chris Thomas, George Martin's assistant. George Harrison would use the mandolin in his post-Beatles career, but as far as I can tell never used the instrument prior to 1970.

There is a very small but interesting parallel between "The Continuing Story of Bungalow Bill" and "Lucy in the Sky with Diamonds": Both songs feature three hard percussion hits that bridge the verse and chorus. Rather than attempting to write out and describe this comparison, it is probably easiest to simply listen for yourself:

A similar effect is used in "Glass Onion", except with only two percussion hits instead of three.