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Photo: Jóhann Heiðar Árnason, CC BY-SA 3.0 via Wikimedia Commons
At the far western end of the peninsula on which Reykjavík sits there is a small island. This island, called Grótta, is connected to the mainland on low tides and open to the public most of the year (closed May 1-Jul 15). Just make sure you don't get stuck on the island when the tide comes in!
Grótta is a tied island at the extreme end of Seltjarnarnes in the Capital Region of Iceland. During high tide the tombolo is completely submerged, turning Grótta into an island.
© OpenStreetMap contributors
The city's oldest outdoor pool. Located in a residential area. Of the pools on this list, this is the only one within a moderate walking distance of the city centre.
Landakotskirkja
The Catholic cathedral of Iceland.
With the exception of Húsavík in the north, Reykjavík is actually one of the very best places to go whale watching in Iceland. Whales frequently come into Faxaflói, the large bay which Reykjavík sits by and on a typical trip of around 3 hours you can almost be guaranteed to see at least some minke whales and possibly even a humpback. The companies offering whale watching mostly occupy a small area in the old harbour called Ægisgarður, close to the whaling ships. All sail out to the same bay but since conditions there change make sure you are on a good ship.
Ljósmyndasafn Reykjavíkur
A very small museum with a nice library and reading room where you can find some older (but good) books about photography and current and past issues of photography magazines. It also has a huge collection of Icelandic photographs.
The Settlement Exhibition
Run by the Reykjavík City Museum, this exhibition in central Reykjavík was built around the oldest archaeological ruins in Iceland. As the name indicates, these ruins date to around the year 870. This interactive exhibitions brings you the early history of the area that today forms central Reykjavík.
By the old harbour in Reykjavík, Hafnarhúsið hosts a rotating exhibitions of the work of Icelandic artist Erró and temporary exhibitions often showcase other local artists.