E.g. canary eggs, parrot eggs, etc.
They do, under the right conditions. They need a nesting box, the right kind of food, and I believe that they need the right lighting conditions, too.
Most laying chickens live their lives out in cages.
Yeah I meant pet birds you get at the pet store.
I’ve read instructions on how to breed parakeets (budgies) and canaries. It’s possible to get them to lay eggs, and to hatch those eggs. I don’t remember the exact details, as I read these instructions over 30 years ago. But it’s possible.
Just google up “breeding parakeets” or “breeding canaries” and you’ll get lots of hits. I imagine that googling other kinds of birds will also provide hits.
Most female birds won’t be stimulated to lay eggs unless the conditions are right: they have to have the right food, a nesting site, and most importantly, a male bird that provides the right courtship stimulus by song and other behavior.
Domestic chickens lay eggs without this stimulus because they have been selected to do so over thousands of years.
Female birds will sometimes lay an infertile egg without stimulus, but that’s not common.
Ah, so chickens are the exception, not the rule?
Cool. 
Very much so. Female domestic chickens will lay readily without the egg being fertilized. As long as the eggs are removed, they will keep laying almost indefinitely, and can produce hundreds of eggs a year.
Anecdote time!
Years ago I had a friend who owned a cockatiel. The bird would occasionally lay an unfertilized egg on the bottom of its cage. It looked like a white peanut M&M rolling around on the bottom bars. If my friend didn’t notice it – say it rolled under the feeding tray – then eventually it would break and stink up the room.
Our cockatiel did this ALL the time too! We’d remove the eggs every few days. She did this for years on end with or without male stimuli. Crazy little bugger. One moment she’d be yearning for a petting when you reached in, the next moment you’d rub one feather the wrong way and she’d bite a finger off.
Yup, we had a one-eyed (!) lovebird* that was an egg-laying fiend, with no male in sight. She even had an “egg song” that she would sing to her eggs. Imagine the chirp your smoke alarm gives when its battery is dead, only every 5 seconds instead of every 30.
The first time she laid an egg, Mr. S thought I put a fake lovebird egg in there as a joke. First, where would I get a “fake” egg the right size? Second, did he not notice the flecks of blood on the thing?
We just took each egg away after a few days.
She got egg-bound once (where the egg gets stuck in the bird’s body) and we had to take her to an avian vet. Years later she got egg-bound again, only we didn’t notice until it was too late. 
She was getting up there in years anyway, but still . . .
- Arrrr, me buckos . . .
My friends used to breed Parrots. It’s difficult to get them to breed- the temp and humidity must be right, the food, the cage must be just so, and so forth. But it can be done.
I used to work at PetCo, and we would occasionally find eggs in different cages. They were unfertilized and if left they would get broken by the other birds playing and make a mess. The reptile specialist would keep them and feed them to the nile monitor.
They’re too busy singing.
(Or killing.)
I know someone that had many baby birds to give away over the years. Her birds had no reproductive problems. It’s not like the mentioned birds lay eggs all the time like chickens. That’s selective breeding that did that to chickens.
My cousin had lovebirds, or finches (forgot which) with a nesting box, and they laid eggs and had baby birds all the time. She had so many birds she was giving them away to interested neighbors! Don’t know how or why this happened, but she didn’t set out to breed the birds, they did it all by themselves.
A friend of mine keeps finches, which breed regularly. She trades baby birds with the pet shop for food and supplies.
I used to keep zebra finches, and the female would lay eggs frequently.
I wasn’t trying to breed them, and was under the impression they wouldn’t breed without very specific conditions, but the pair I had did manage to hatch 2 eggs.
One died, the other was hatched on election day in 1992, and thus dubbed “Bill”, and lived to a ripe old age. For a finch.
A friend of mine used to breed and hand rise cockatiels and sell them to a particular bird oriented pet store. I had one of her 'tiels named Baby that was amazingly sweet. I do remember the cage for that particular pair had a nest box, and its own light, and she fed them a different diet, it included eggshells for the calcium suppliment as apparently birds other than chickens will eat the leftover shells after hatching.
When I had a pair of lovebirds they raised one batch of eggs through chicks to adulthood (4 babies altogether) and the female laid a couple more eggs some time after that, but the silly twist used too big a box for nesting (she appropriated them on her own - the first was a shoebox, the next one a box originally for Xerox copy paper. We had hidden the shoeboxes hoping to avoid another round of babies and it didn’t work. What’s this about needing precise conditions for breeding?) and she lost them for awhile. I still have one, very much dried out by now I’m sure. So “pet store birds” do lay eggs
Currently we have two sexually frustrated boy cockatiels and thus do not have to worry about eggs and babies from them. We also have a green cheek conure but she’s not yet old enough to be producing eggs, though she’s getting close. We think she’s female, but being a bird, it’s hard to tell until she actually does lay an egg. Not that we really care either way.
My parakeet randomly laying eggs was how I discovered:
a) He was a She
and
b) The “blue skin above the beak means boy, brown skin above the beak means girl” thing isn’t 100% accurate.