Formal Structure of [129] "Everybody's Got Something to Hide Except for Me and My Monkey":
          Intro (verse)     0:00-0:08
          Verse 1 (abbr.)   0:08-0:40
                    Part 1 (abbr.)     0:08-0:20
                    Part 2                    0:20-0:32
                    Part 3: Tag          0:32-0:40
          Verse 2                0:40-1:20
                    Part 1                    0:40-1:00
                    Part 2                    1:00-1:12
                    Part 3: Tag          1:12-1:20
          Verse 3                1:20-2:00
                    Part 1                    1:20-1:40
                    Part 2                    1:40-1:52
                    Part 3: Tag          1:52-2:00
          Coda (verse)     2:00-2:24

Comments: Many of the songs released on "The White Album" were written as fragments during the Beatles stay in Rishikesh, India in the early months of 1968. 
"Everybody's Got Something to Hide Except for Me and My Monkey" is one such example - its use of only verses (no choruses, no middle 8s, no solos, et cetera) revealing its fragmentary nature. (NOTE: I count the third and final sub-section of each verse as a tag, but it's clearly a sub-section of the verse and NOT an independent structural component due to its brevity and lack of structural weight.)

While it is rather common to find a Beatles song in which the first two verses are contiguous (precedents include [1] "Love Me Do", [7] "Do You Want to Know a Secret", [8] "Misery", [9b] "Anna (Go To Him)", [9c] "Boys", [9d] "Chains", [9f] Twist and Shout, [10] "From Me To You", [13e] "Till There Was You", [17] "Little Child", [19] "Not a Second Time", [23] "Can't Buy Me Love", [25] "And I Love Her", [26] "I Should Have Known Better", [28] "If I Fell'', [29] "I'm Happy Just to Dance With You", [31] "A Hard Day's Night", [31b] "Matchbox", [32] "I'll Cry Instead", [35] "Things We Said Today", [40] "I Don't Want To Spoil the Party", [41] "What You're Doing", [42] "No Reply", [43] "Eight Days a Week", [44] "She's a Woman", [44b] "Kansas City/Hey Hey Hey Hey", [46d] "Words of Love", [47] "Ticket to Ride", [49] "I Need You", [50] "Yes It Is", [51] "The Night Before", [52] "You Like Me Too Much", [54] "Tell Me What You See", 56b] "Dizzy Miss Lizzy", [56c] "Bad Boy", [57] "I've Just Seen a Face", [59] "Yesterday", [66] "If I Needed Someone", [68] "We Can Work it Out", [71] "Michelle", [77] "Tomorrow Never Knows", [80] "Paperback Writer", [82] "Doctor Robert", [84] "Taxman", [88] "Yellow Submarine", [89] "I Want To Tell You", [92] "She Said She Said", [95] "Penny Lane", [96] "A Day in the Life", [99] "Fixing a Hole", [100] "Only a Northern Song", [101] "Being for the Benefit of Mr. Kite", [105] "Within You Without You", [111] "All Together Now", [114] "All You Need Is Love", [116] "I Am the Walrus", [122] "Lady Madonna", [126] "Don't Pass Me By", and [128] "Blackbird"), it is uncommon to find any other contiguous verses. In fact, "Monkey" is just the 8th Beatles song to use contiguous verses other than verses 1 and 2, following [19] "Not a Second Time" (in which verses 1 and 2 are contiguous, as are verses 3 and 4), [31b] "Matchbox" (in which the first three and last two verses are contiguous), [56b] "Dizzy Miss Lizzy" (in which verses 1-2 are contiguous, as are verses 4-5), [77] "Tomorrow Never Knows" (in which verses 1-3 are contiguous, as are verses 4-7), [80] "Paperback Writer" (in which verses 1-2 are contiguous, as are verses 3-4), [84] "Taxman" (in which verses 1-2 are contiguous, as are verses 3-4), [95] "Penny Lane" (in which verses 1-2 are contiguous, as are verses 4-5), and [96] "A Day in the Life" (in which verses 1-3 are contiguous).

With such a fragmentary and minimal macro-structure, it is no surprise to find that each verse contains three significant sub-sections: first, that corresponding with the lyrics "Come on, come on..."; second, that corresponding with the lyrics "Take it easy..."; and third, that corresponding with the brief instrumental break immediately following the title lyrics.

Additionally, notice that the first part of verse 1 is abbreviated, lasting only 12 seconds - 8 seconds shorter than the first parts of both verses 2 and 3. The following side-by-side comparisons of the three verses show this abbreviation visually (click to enlarge):

By observing this graphic, you can see that verses 2 and 3 are practically identical, but verse 1 is shortened, missing the first six measures of the other two verses and substituting two measures of comparable melodic material.

Lastly, also notice that Part 1 of the first verse concludes with a 9/8 bar while the other two verses retain 4/4. As to why Lennon decided to make only the first verse this way, I am uncertain.

[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: