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Reykjavik seems expensive to many people, especially when you look at the costs of things such as eating at cafés, buying clothing, getting haircuts at a salon, and buying anything that had to be imported (which turns out to be most things). It’s good to know where you can get free stuff, especially considering free things are nowhere near as prominent here as in America.
As for drink, Íslandsbanki right next to the pond offers free coffee from a machine just inside its doors. Some clinic waiting rooms also have free coffee machines, but hopefully you won’t be visiting those often. Water fountains are scarce, although one does exist on the main shopping street and more exist in the University of Iceland’s buildings.
For food, occasionally sample stations do exist inside grocery stores but I’ve yet to see any regular schedule. At least at my grocery store, they’re usually for baked goods that seem rather dubious. Sometimes on the weekends there’s such things as free soup in Lækjartorg (a small town square right in the middle of downtown), and while I was told that’s usually done by Hitt Húsið I haven’t often checked it out so I’m not sure. Depending on your workplace they may also give you free lunches as part of your contract, if they have some sort of kitchen. At the flea market you can try a little piece of hákarl (a type of Greenlandic shark – in this case meaning it’s been fermented too) for free from one of the booths that sell it, even if you’re Icelandic.
Most commonly around Christmastime the post office or bank may have cookies or candy. Shops might also have this, and usually they’re some type of gingerbread cookie. Immediately before Christmas there may be some little stands with free hot chocolate or coffee downtown, mostly on the main shopping street. I think it’s pretty rare to see this sort of thing outside of that season.
The post offices all have packing tape and scissors for free use. In some places, for example inside Kringlan (a mall), Eymundsson (a bookstore chain) and Toys R Us, they have wrapping paper for free all year round too.
Some cafés, parks, and public squares have free wireless internet, but it’s not actually common and you shouldn’t expect any businesses to have it. If you want to exchange money you should most certainly do it at a bank, where it almost never costs anything – if you do it at the airport or any other sort of place they’ll charge you a fee, and you might even get a worse exchange rate.
At the main town library there is a “green books” (“economic/recycled” seems to be the meaning of green there) table that has books you can borrow without using a library card, although from what I’ve noticed most people never return the books. Sometimes they also have a “take a book, leave a book” set-up for people’s own used books.
(This is the room described below.)
At the University of Iceland in one building is a table where people place giveaway books, lost books, and leaflets, and in the same room they pin up things such as lost mittens to the corkboards. You’ll often see scarves, shirts, and mittens tied to poles or caught on fences around town after a day of strong wind, so their owners can find them again later. If you happen to live in Reykjavik then you can keep an eye on these places, I once noticed that a certain set of mittens stayed on the corkboard for an entire year so by that point it’d probably have been safe to say the original owner wasn’t going to find them again.
The Nordic House very often has free events, sometimes including brunches. The events vary widely between live music, plays for children, poetry reading, movie showings, and academic discussions. They’re also not necessarily in English or Icelandic. Sometimes they have old flyers, movie posters, and newspapers that they give away. The University of Iceland sometimes has free events similar to the ones at the Nordic House, although those are usually in Icelandic and secondarily in English, and the Reykjavik town library sometimes holds events as well. One weekly thing at the town library is that you can go there to get help reading Icelandic newspapers if you’re a foreigner, if I find a photo of the advertisement I’ll edit this post to include it.
Other free events include some things listed at Visit Reykjavik. For a while there was something called “Couch Fest Films” being advertised, where you visited a stranger’s house to watch a movie with them and some other strangers. Sometimes art galleries have free showings and there are also lots of free concerts around too, although I don’t go to such things so I don’t know how you’d specifically look for them.
Many places such as the University of Iceland, some cafés, embassies, and the hospital have flyers and maps that you can take for free. Some are in English but most are in Icelandic. If you want to get something random in Icelandic to practice with or remember your trip by, it’s much cheaper to take a bunch of these than to buy some books.
These two things are probably only very useful to people living in Iceland, but you can text Icelandic numbers for free on ja.is (excluding Nova users, I think it was) and you can recycle bottles/cans to get money. I’m sure you initially pay a deposit on those bottles when you first buy the drinks, but especially if you have friends who are too lazy to recycle, maybe you can make some kind of profit.
Lastly, hitch-hiking is said to be extremely safe in Iceland. I’ve had some friends who spent even months just hitch-hiking around the country, and they said at most there was one or two times where things seemed a little strange but even then nothing actually happened. Overall they had a great time and met many great people. I’ve only hitched rides for very short distances, and as with all things there’s a chance of danger no matter how safe things seem, so I can’t say much about it.