2 or 3, depending on adjacency. 3 can be nice to get a discounted 1st theater, but you really only need +9 adjacency and you can often get that with 2 campuses. You need to find a way to get science to 130/turn or so by the end of the game. You don't need most of the lower part of the tech tree, but you need to get to flight (too many tourism-generating things to spam these days), then radio, then computers. Radio and computers aren't really needed in low-opponent culture games, either. The rest of the tech tree doesn't really matter much, unless you are playing a really long game (have Peter generating a lot of culture, for example).
In a typical game for me these days, I am getting more tourism from builder-built things than from great works. Hmmm, that gives me an idea for a challenge. I think I'll start another thread.