Thanks for all the suggestions, everyone!
Interesting to see how many different favorites people have (and how many of them I haven't tried). I'll have to do some more research and decide what appeals to me most. I think I'll make a big list of all the suggestions and examine them one by one.
I'm probably leaning against RTS-type games. I've played some I liked, but I generally find them tiresome if I have to control each unit individually. Just a matter of taste.
I'm leaning toward games that still have active modding communities. That's always a good sign.