Apart from the usual electrical bodges found on almost all second hand cars, the two worst bodges I've come across after buying a car were:-
On a Mini, a previous owner had fitted one of those small bracket things to drop the steering wheel which wasn't the real bodge, they'd fitted it without adjusting the steering rack to allow for the different angle of the steering column. That put extra strain on the spline joint between the rack and the column which, you've guessed it gave way one day. Not a pleasant experience trying to brake without being in control of your direction. As they say, "Squeaky bum time".
The other bodge I came across was on my old Lotus. For a long time I couldn't work out why the engine looked like it was on a tilt. As you're probably aware the Elan was a fibreglass body on a separate chassis and it turned out that the front body mountings had collapsed at some time. That dropped the body onto the engine, well, the airbox on the carbs to be accurate, so rather than repair the body mounts, the previous bodger put a spacer under one side of the engine mounts to tilt the engine so that it cleared the bonnet when it was closed. That one had me scratching my head for quite a while.