Wel, when you're talking stat checks, the kind of activities are necessarily limited by the kinds of stats we can check against? Is it the standard D&D STR/DEX/WIS/CON/INT/CHA? A Fallout-Esque S.P.E.C.I.A.L.? An early-Final Fantasy Strength/Intelligence/Agility split?
You tell me what kind of stats there are and I'll be able to come up with all kinds of scenarios and events for them.