Having just this past September come across an '06 GSX-R600 on Craigslist (for $900! It was a mess with a dubious engine to boot), the Missus reminded us of the cardinal rule of used motorcycle buying:
Probably a chain. Another $150. While I've never had one of my own chains go less than 40K, not many people do chain maintenance. Her SV650S's chain was ruined at only 5k- the guy said "I didn't know you had to lube a chain."
What happens when a chain fails at speed is pretty terrifying.
I had trouble on our 600 with the cam position sensor and the cooling system, requiring a thermostat and radiator cap. Again, maintenance done wrong or not at all.
But tires. Not everyone knows that tires have a service life of only five years before they harden and lose performance. That especially goes for bikes stored outdoors. There's a date code on the sidewall in an oval showing the week and year of manufacture. Personally, I won't go past six years even on nice-looking tires.
I recently got an old Concours for a project. The tires were dated 3 1/2 years ago. But the bike was parked outside the whole time. The rear tire, especially, feels like it's been Vaselined. Scary.