I attended an Anish Kapoor exhibit here in 2010 that was mind blowing. The exhibit was based around huge volumes of wax. In one room, there were canons that shot the crimson colored wax against a wall, and it then fell onto a huge pile. In another room there was huge blocks of wax on conveyor belts. This isn't saying much about the Royal Academy or Arts itself, but what it does say is that the place is home to some mind blowing art exhibits. I imagine that most of what they host is worth your time. The old buildings themselves are interesting enough to look at as well.
Huge, historic, gorgeous building that houses a permanent collection and some great exhibitions.
Cannot wait for up coming Hockney exhibit.