Fridge Brilliance: It didn't make sense that Remy's plan to get out of the system went without a hitch. The two main plot holes were when Jake just let Remy and Beth go after the airport fight scene, and when Remy survived a fight largely outnumbered outside the pink door. The answer is that Remy was hooked up to the M5 Neural Net.
The airport fight scene was before Remy was hooked up to the neural net.
Why would the sign in Union headquarters say "The Pink Door?" It's because that's the only name Remy knows.
How did Jake know about Remy & Peter's talk about the Romans? Because Jake was Remy's sockpuppet.
There are several clues throughout the movie that not everything is how it seems. The most obvious one comes at the previously mentioned fight outside of the pink door, where Remy tears through a bunch of businessmen and people in street clothes, all of whom are wielding a series of bladed weapons(such as a meat cleaver!) that they really had no business carrying around, even if they were repo-men. The second is that Remy only knows of four instances of losing consciousness, when he actually gets knocked out five times. The fifth knockout is what gets him hooked up to the Neural Net.
Fridge Logic: It seems weird that, after living in the States since at least 3rd grade, Remy would still have an English accent.
It's surprising that there's no spike in crime, or mob takeovers. With all the families unable to pay for their organs, you'd think they'd be more desperate than just trying to hide.
The business probably have such political power that they make sure none of their clients even try.
The explosion in the seemingly air-tight sealed room behind the Pink Door nearly broke this troper's Willing Suspensionof Disbelief due to the oxygen consumption and smoke/fumes required for such a fire being ignored. The Fridge Brilliance did restore it slightly.
Is there no such thing as workman's comp in this setting? Because going into cardiac arrest due to faulty equipment is a fairly legitimate claim, not to mention the good publicity the Union would generate by fitting one of their top "Repo Men" with an artificial heart for free because of a workplace injury.
Not to mention the fact that Jake was responsible for the accident. Remy would have an excellent case if he went to court; Jake could be held liable for Remy's medical bills.
Fridge Horror: Before repo-ing, their organs out of the system, Remy and Beth take a hit of a drug to help dull the pain of this. This is fine for Remy, who has no stated history of using the drug. Beth, on the other hand, was an addict, and one hit would hardly affect her