These are a few items I've read about:
Some have been getting hot starting problems, the cause as often as not has been the starter motor, usually with a mileage near to 100000 miles. I suspect it depends more on start cycles than mileage, but it's rare enough not to have statistics.
Alternators (and the associated RR) seem to last indefinitely, with a very few random failures. After-market alternators are definitely less reliable.
Water pumps - a very few random failures, usually starting to show up as minor leakage, so giving plenty of warning.
There are some electrical failures, the most significant was the so-called "spider" failures on GenII's ('06 - '12), where corrosion has caused burning of contacts on the (somewhat strange) earthing arrangements. Also some ignition switch failures on GenI's and GenII's. Some of these have been covered by a Yamaha recall. Battery life varies, no particular reason why a few only last 2 years, while some last 8+ years.
You'll find this bike most unlikely to give you any problems provided you service it properly (plus a few little extras that have been learned by the forumites here over the years).
Personally, the only "failure" I've had was a sticking clutch issue, resolvable with a couple of hours labour, never going to leave you standing by the road.
I've never felt any anxiety about the bike's reliability however far from home I'm going (ok, I've not done more than 40000 on any of my FJRs before I've changed, but that's nothing to do with fear of breakdown).
