Favorite thing: Take a trip outside the city to see Gullfoss, the 'Golden Waterfall'. Even if you're just visiting the city for a couple days on a layover, make an effort to get outside the city to see the countryside. The trip to Gullfoss takes about 3 hours each way and passes through some spectacular scenery. The waterfall itself is breathtaking, particularly in the winter, when every exposed surface is covered with cascading sheets of ice. Also nearby is Geysir, a spouting hot spring that gave its name to geysers worldwide. The original Geysir itself is now dormant, but nearby Strokkur reliably erupts every 5-10 minutes (apparently the world's most active geyser).
The Geysers which are not in Yellowstone are in Iceland. They have invented the word : GEYSIR.
Those I saw are smaller than the American ones but can be approached closer, the soil did not seem to be dangerous.
Favorite thing: Take the Golden Circle tour. This is the most popular excursion out of Reykjavik and for a reason. It takes 4-6 hours and an Icelandic guide gives you the inside info on Geysir, Gullfoss (the golden waterfall) and Þingvellir, the site of the world's first parliament.
Favorite thing: gulfoss is amazing. In winter it is frozen like this. In summer you can see the rainbows dance off the water,