If you want to do something truly unique and adventurous during your stay in Budapest, make a trip to the 30 km-long Pálvölgyi Caves. Getting there is really simple: just take the 9 bus to Kolosy Square (you can also walk there in 25 minutes from the city center, as Budapest is not so big), and then catch the 65 or 65A bus. The 5th stop is directly in front of the caves, and the name is "Pal-volgyi-barlang/cave" so it's impossible to miss!
Make sure you call/email ahead to book the adventure tour - there is a 45 minute walking tour but it's definitely not the one you're looking for. The 2.5-3 hour adventure tour will take you on a 1 km loop of the caves, crawling, crouching, sliding and climbing all while your guide teaches you about the formation and history of this beautiful subterranean system. Speaking of guides, ours was awesome. He was really knowledgeable, and an incredibly strong caver.
Be warned: this is not an adventure for the claustrophobia-inclined. There are some seriously tight stretches, and it's an adventurous workout the entire time (except during the rest stops). As long as you're okay with that, you'll absolutely love this adventure. It definitely is not the kind of thing you expect to find in a touristy European capital. Good luck!
Location:2nd district, Szépvölgyi street 162, Budapest, Hungary