vBulletin!
I run a few very elaborate websites. I've been able to do everything I need to do with free open-source software/libraries, custom PHP apps, or a combination thereof--EXCEPT for forum software. vB is much, much better than anything else around. Not just in terms of the software itself, but also the 3rd party addons and enhancements that are so abundant. You can get a lifetime license, or a yearly one for about $90/yr.. You don't have to renew it to keep running the forum either, but you won't be able to get any more updates.
I used to run ikonBoard. It was OK, with the major exception of a security hole causing our site to be hacked! They fixed it, but I still never really trusted it after that.