I got 4 events in my current game that were big time. I messed up on one of them though.
Background: I was Incan, wound up on a crappy island for an isolated start. The good news was that there was a long river = floodplains on the west side, the bad news was that a giant portion of my island was desert, and the east side was a big jungle that took forever to hack into submission.
1. library quest. I was going to try to tech and build like crazy, and as usual, I got the Great Library right after receiving this quest. Don't bother if you don't want to, because all that gives you is the option for a 3rd sci from the library. At the speed that I teched, I lost that sci pretty quickly (EDIT: or maybe I just miscounted? someone below says his stayed on.. one of us miscounted); I should have taken the "all libraries get +2 beakers" option instead. Assuming that it's permanent. At least that sci helped pop out 2 Great Scientists; I made 2 academies. It's actually more effective to do that if you have 2 mass-cottaged cities, than to Academy + meld. No need to bulb as I had a big tech lead after a while.
2. forges quest. This was bizarre because I had six forges and a seventh was about to get pop-rushed when I got this quest. After some debate, I took the free engineer option. Thankfully that engineer doesn't seem to expire. Cottage-spamming's main drawback = lack of hammers so this was a welcome change.
3. spices showed up. The big jungle had mostly grasslands, some hills, and a bit of food, so I was sad to replace a coveted hammer-producing forest tile with spice. On the other hand, I had to do it, I had only sugar for Happy and was dying without more Happy resources.
4. silver was discovered. I had one city way down south near the pole as my designated national park city. It was next to deer, iron, tundra, ice, and a few forests that I hoped would grow. The silver that showed up next door made it all worthwhile. Needless to say, I paid extra gold to have that silver be mined+roaded on the same turn. (My workers were all busy; that jungle took FOREVER to chop down.)
I also had some miscellaneous bad stuff happen to me like farms getting whacked but that's fine. The good stuff can be permanent (see above), whereas the bad stuff does not seem to ever be permanent (not if you have gold anyway, and I founded 3 religions, all shrined, and 2 were in one city that was declared my Wall St city), except for diplo stuff that might fade over time anyway. And I was isolated so I had no diplo random events. (Aside from the tech drawbacks of isolated starts, it sucks too for spies... I had lots of Great Spies courtesy of some wonders and nobody to build up those spy points against.)
I'm on Monarch/fractal/standard and currently in 2nd place only to one guy who vassalized two other civs and is several techs behind me. I went uber cottage spam and hope to get a battleship fleet to ward off any would-be invaders. In the meantime I have just a couple of chariots and a bunch of quechas (warriors) defending.
I infected my similarly-isolated neighbor Brennus with my state religion so hopefully I have ONE guy I don't need to worry about.