BuddyPress.org

If it’s a cheap shared host it may fail if membership is high. I use media temple shared and for small networks I haven’t had a problem. If you have significant traffic you should look into dedicated hosting.

The main difference of cheap hosting to media temple that i’ve experienced is they don’t shut your site down for high server loads they just add onto your bill. I had a test BP site on a cheap shared host and about 300 people tried to log in at the same time it tipped the server and they shut down the site for 3 days.