Perlan, Reykjavík Region
The building on the hill is The Pearl (Perlan), an exhibition and conference hall with revolving restaurant on top, well worth visiting. It is built on top of old water cisterns and the view from there is lovely. But the hill, Öskjuhlíð, is a natural pearl in its own right. More than 176,000 trees have been planted on the hillside. There are bicycling and walking paths and the hillside is frequented by locals and visitors alike. This peaceful place is especially charming in the summer.
The Pearl is quite an unusual structure. A restaurant is found housed on top of these large tanks that hold (or held) geothermally heated water. The restaurant here is very expensive so we didn't eat there but we did go to the observation deck and had quite a lovely view of Reykjavik. It's not in the city center but worth the trip. You can also find the remenants of a dormant geyser right outside the Pearl. Indoors, they have a man-made geyser that seems to erupt every 2-3 minutes. It shoots up quite high.