Pálvölgyi Cave

How about exploring the capital of Hungary with a café during the AM and a cave during PM? 

The dripstone-rich cave, the longest one in the Buda Hills and the third longest in the country, is a highly protected natural preservation area since 1944.

Its length exceeds 7,200 meters; the vertical dimension is 104 m.

The temperature is a constant 11 Celsius, so dress up well, bring hiking boots and expect to climb ladders.

Check out this 36 second-video (linked from Youtube) if you consider going:
Pálvölgyi Cave video

Prices are from 1200 HUF pr person.