What non-metric fastener heads are left on a standard bicycle?
Very rare, but I believe some pedals still require a 9/16" spanner to remove them. Not to be confused with the 9/16" axle like you have mentioned. Park Tool still make a pedal spanner with both a 15mm and 9/16" opening. Considering 9/16" of an inch is 14.3mm and the chances you'll actually have to remove one are fairly slim, I'd be happy with just a standard 15mm pedal spanner. 

Beyond this I can't think of anything else. Perhaps wheel nuts on very cheap kids character bikes? Some of those seem to come with whatever the factory had going spare at the time of assembly!