What seems to be the most common problem? This is the 4.4 V8
Engine wise:
-timing chain guides (plastic lined aluminum guide, chucks the plastic off to be chewed up by rotating parts, then makes a gaggle of noises)
-secondary air ports becoming clogged in the cyl heads (can require removal of cyl heads if bad enough)
-engine oil leaks. I know, Bmw, to be expected...... but these oil leaks are next level. 3 gaskets on the oil pan, front cover, valve cover etc.
-cam followers wear out
-central bolts for cam gears back out (rare occurrence, but have seen it happen twice that I can remember).
-some models were equipped with a water cooled alternator. Not bad to repair if it goes, but really expensive part.
-oil separator vents the majority of the pcv gasses into cyl 7 or 8 (can't remember which) Can cause washing of cylinder walls and potentially cylinder damage.
E53 chassis:
-ball joints, all of them. A lot. If it has rear air springs, deflate air springs to have the rear suspension checked. If it isn't deflated, suspension will check ok, until one of the ball joints falls out.
-cv shaft boots split open, inner and outer on front axles
-Brake lines, to the rear, in the area of the drivers feet below floor boards. (Probably different for southern climates)
There could be some stuff I missed.