Acetaminophen is the safest painkiller to have on a mountain bike ride (don't take more than 3000mg a day. Acetaminophen is in in part metabolized by an enzyme that prevents free radical oxidation as long as you make enough of that enzyme a day, which a healthy liver does, then you won't have liver toxicity issues). The risk of damage from acetaminophen in one day or just a few days of use is very low provided you stay below the recommended dosing. people get into trouble thinking it's a "harmless or safe medication" and often taking more than the prescribed limit or unbeknownst to them, having a liver enzyme deficiency (that's why personalized medicine is the next big thing.)

Ibuprofen (an NSAID, like aspirin or aleve) on the other hand, increases the time needed for blood to clot resulting in greater risk for bleeding in the brain/around the brain or in other areas after a possible unexpected crash. As a result, I've seen people taking aspirin (its effects sticks around for one week after taking it) and ibuprofen (sticks around for about two days after taking it) with severe bleeding in the brain after relatively minor falls. Additionally, ibuprofen decreases the amount of blood that gets to the kidneys causing a trifecta of badness if you're dehydrated or have elevated muscle breakdown products ( say from a long ride) etc. Lastly, ibuprofen but notably not acetaminophen, have been associated with higher risk for cardiovascular events including heart attack and stroke in the long run. The safest NSAID, at least for those complications, is probably naproxen.

Sorry for the long post I just wanted to clarify a few things.