Executive Meddling: After the Beasties fled to Capitol, Def Jam head Russell Simmons announced that he would assemble an album created from the band's outtakes with extra contributions by others (think Tupac Shakur's and The Notorious BIG's posthumous albums). Paul's Boutique came out first, and nothing was heard about Simmons' idea again.
Old Shame: According to the Beasties themselves, Licensed to Ill. Their early EPs and singles also fit into this category, given that they eventually re-released them on a compilation entitled Some Old Bullshitnote They seem to at least have some affection for their early song "Egg Raid On Mojo" though - they included a lyrical reference to it in "Egg Man", occasionally played it live even after becoming primarily a hip-hop group, and it's one of two songs from this period to appear on the band-curated compilation Beastie Boys Anthology: The Sounds of Science (the other one being their namesake song). They also regretted their party-boy antics when they hit it big, and cut out misogyny in their later albums.
One of Us: The Giant Robot-Space Monster battle in the video for "Intergalactic". They also later changed the official lyrics of "Ch-Check it Out" to use the Klingon reference from the clean version, and the music video is one giant one for Star Trek. For more, check the Shout-Out section in the Music tab.
The well-known example of Rhyming with Itself in "Pass The Mic": Mike D meant to rhyme "commercial" and "rehearsal", but in the take they used on the album, he accidentally repeated himself, resulting in "Everybody's rapping like it's a commercial / Acting like life is a big commercial".
The Pete Best: John Berry (guitar) and Kate Schellenbach (drums) left before the band signed with Def Jam and switched from punk to rap.
What Could Have Been: In interviews, Mike D mentioned the idea of releasing a With Lyrics version of The Mix-Up as a companion to the instrumental one, with guest vocalists on every song. MIA and Jarvis Cocker were rumored to be among the contributors.