merapi (+/- 2900 m) is one of the most active volcano in the world. it's dangerous, but also one of most beautiful place in yogyakarta.
you can enjoy a lot of adventure activity here, like hiking, rock climbing, bird watching, camping, etc.
there's also a lot of cultural attraction too. merapi is a place where myth and supernatural stuff still strongly belief by the javanese specially who live around merapi. even the eruption is considered as a gift, not disaster. it has strong connection with keraton Sultan, king of Jogjakarta.
you want something adventorous? you can get it there, it's a beautiful mountain.
you want something cultural? you'll get it. there's Ulen Sentalu, a beautiful Jawa museum in Kaliurang, in southern Merapi, villages with strong Javanese culture, good Javanese food (find Jadah Tempe Bacem if u're there), and of course, interesting story myth. :-)

Along with Mt Bromo, this is one of the best active volcanoes to get up close to in Indonesia without too much effort, (compared to Gunung Rinjani in Lombok!). The night we were there it was erupting great fireballs of molten rock and lava down it's slopes, which was awe inspiring. Make sure you take a jacket as it gets pretty cold up there, and if you're into photography, bring a tripod for your camera.