Front suspension arms. Fairly expensive to replace and very annoying because the whole arm needs replacing for the sake of a rubber bush that would cost a few quid. After replacing their initial nice, tight and new feeling deteriorates within a couple of months.
Rear suspension bushes. Sounds simple again but they are pressed into the chassis of the car & unremovable. Fiat realise this design mistake and sell a rubber cup that fits over the original bush to try and patch up the problem.
Clutch replaced 4 times (3 times by a Fiat dealer - never again). Turned out to be dowels missing in the gearbox causing the problem.
Exhaust mount has fallen off (welded onto to the chassis). Resulted in a split in the braided hose on the downpipe, again very expensive caused by a part that costs pennies.
Fiat's infamous Keycode system. Occasionally the car decides that the key you've been using for the last couple of years is not the right code thus disabling the ignition system.