There aren't very many plants which will actually bloom year-round, but there are a few, and there are even more than that with long bloom periods that might as well be continuous bloomers. As always, whether or not you actually see continuous blooming from one of these plants in your home is going to depend on whether or not you're providing the care it needs.
It's a short list this time, because honestly, I had a lot of trouble coming up with much. (It's a valuable characteristic because not many plants have it.) So by all means, if you know of another plant that belongs on the list that I haven't included, leave a note in the comments.
We had Acalypha reptans most of the time when I worked at the garden center -- it wasn't an incredibly strong seller -- and I didn't like it very much, because it was messy, but I don't remember seeing them without flowers very often.
Anthurium cvv. (shown: NOID) I have personally had a single Anthurium plant in bloom for two years straight before. This is at least partly because Anthurium flowers are very long lived: two to three months, on many varieties.
I haven't personally grown a Bougainvillea (shown: NOID) indoors ever, but I'm told that they can bloom continuously in at least some circumstances, and people do grow them indoors, so I think they technically count.
Euphorbia milii (shown: NOID) does seem to go through occasional brief rest periods for me, but that might have more to do with our weather than with day length. Certainly it blooms a lot of the time even if it does take breaks occasionally.
Hoya lacunosa blooming happens in waves for me: it produces a bunch of flowers about a week after a watering, then takes a break for a week, then flowers again. It's the only Hoya I have that's actually been willing to flower for me, but does so really easily.
Justicia scheidweileri (also still known as Porphyrocoma pohliana and a few other things) seems to be blooming much harder for me now that the days are longer, but it's at least had visible bracts, if not the actual flowers, for most of the last year. It did take a short break in the winter.
Murraya paniculata is another wave-type bloomer, with flower buds appearing in huge numbers about a week after a watering, then opening, drying up, and falling off a few days later. The smell can sometimes be intense. This photo shows only the one flower, but this particular plant can easily produce 50 at a time; I'd apparently caught it at a bad moment.
My personal Saintpaulias (shown: NOID) -- with one exception -- are all ungrateful assholes who haven't bloomed in about four months now, and seem determined to make me furious with them. (The exception is a plant I've had for a long time, which has truly never been completely without flowers in at least a couple years.) See below for theories. But other people, not me, get them to bloom all the time.
Spathiphyllum cvv. (shown: NOID) do have something of a bloom season (roughly February to October), but in good conditions, they can form blooms at any time.
The only two on this list that I haven't tried to grow are the Acalypha and the Bougainvillea, so I can't really speak to how easy or difficult they might be. Of the others, my three recommends would be Anthurium cvv., Euphorbia milii, and Hoya lacunosa, with Abutilon as an honorable mention.1
Anthurium cvv. and I have been friends for a very long time now. I grow mine in a large, east-facing window, so they get morning sun, though the light is somewhat filtered by other plants. Soil choice is very important -- the plant in the photo above was killed by a badly-timed repotting into soil that was too heavy. (I recommend cutting a regular, good potting mix about 50-50 with coarse, unchopped sphagnum, though smaller plants in smaller pots can usually be potted into unamended potting mix.) It's been my experience that keeping them too wet is more likely to be fatal than letting them get too dry, though in an appropriate potting mix you have more leeway than you'd think. A lot of people say they need high humidity and warmth; I haven't found humidity to be that critical with the varieties I have, but temperature is still important -- mine are at about 70-75F / 21-24C, round the clock. They'll bloom without much fertilizer, but blooms are larger and more abundant when they're well-fed. Flowers are unscented.
I had my Euphorbia milii for over a year before figuring out how to convince it to bloom: it turned out to need more fertilizer and light than I was giving it. They're a little more difficult than the average succulent Euphorbia, in that they prefer to be a little wetter than you'd expect, but they're very tolerant of low humidity and fairly resistant to pests. (Mealybugs are always a problem.) The sap is poisonous, as is the case for most/all succulent Euphorbias: definitely avoid contact with eyes, keep away from children/pets, the usual drill. Flowers are unscented.
Hoya lacunosa isn't much to look at, but it flowers freely in bright light (mine didn't start to bloom until I had it almost touching a pair of fluorescent shop lights -- the leaves began to yellow and everything -- but once started, it's never really stopped, even though I moved it away from the shop lights ages ago), if kept well-fed. Some direct sun helps. The flowers have a strong pleasant scent, which to me smells like a florist's display case.
For the traditional anti-recommend, I'm torn between Saintpaulia and Bougainvillea, but I think I have to go with Saintpaulia.
Bougainvillea I mostly dislike because they get big and awkward to move around, plus they have large, sharp thorns. This is why I've never tried to grow one.
Saintpaulia and I, on the other hand, have tried over and over, and with a single exception, it's never gone well. Usually I wind up underwatering at some point and they die, but currently I think the problem might be cyclamen mites. All but one of my Saintpaulias stopped blooming a few months ago; several have been producing stunted, curled or twisted leaves; and one of the ones with the twisted leaves also has weird tan patches on the newest leaves. This could be an indication of cold damage, which a couple people have suggested to me by e-mail, and they were in the (cold) plant room all winter so that's plausible, but I moved all of them into the (warmer) living room a few weeks ago, and they're acting the same or getting worse. I'll try to get some photos up sometime this week, in hopes that someone can confirm or rule out the cyclamen mite theory.
This is not typical of most people's experiences with African violets, and it wasn't typical of my experiences until about February, but I'm finding it very hard to come up with anything nice to say about them at the moment.
- Begonia cvv. (wax begonias, maybe also Rieger begonias and non-stop begonias)? I saw wax begonias mentioned as constant bloomers on a website or two while I was writing this; I don't know if it's actually true when they're grown indoors.
- Brugmansia cvv. (reader suggestion)
- Codonanthe cvv. (some cvv.) (reader suggestion)
- Columnea cvv. (some cvv.) (reader suggestion)
- Episcia cvv., at least some cvv. (flame violet) My Episcia 'Coco' has been in nearly-constant bloom since I got it last summer. It's been under artificial light, so it's possible it's confused about what time of the year it is. None of my other Episcias are anywhere near as consistent as 'Coco' is, though.
- Fuchsia cvv. (reader suggestion)
- Hibiscus rosa-sinensis cvv. (tropical hibiscus) The ones we overwintered in the greenhouse at work were more or less constant bloomers, though they did slow down considerably in the winter. I'm presently finding it tough to say anything nice about Hibiscus as well. I almost decided not to try to overwinter them last year, because they'd gotten spider mites bad the year before, but I went ahead and saved them anyway. They didn't get mites, but large sections of both plants died; I should have gone with my first impulse and chucked them.
- Hoya bella (reader suggestion)
- Impatiens cvv. (some cvv.) (reader suggestion)
- Kohleria cvv. (some? all?) Long bloom period but not, strictly speaking, ever-blooming. (reader suggestion)
- Mammillaria rhodantha (reader suggestion)
- Mandevilla cvv. Another one that bloomed continuously in the greenhouse at work, but might or might not in the home. They're so prone to spider mites that I would never bring one into my house on purpose, though.
- Osmanthus fragrans (sweet olive) (reader suggestion)
- Oxalis cvv. (some cvv.)? Another one I saw on websites but have no direct personal experience with. I know for certain that not all Oxalis varieties bloom continuously.
- Pelargonium cvv. (geranium) Another one that bloomed continuously in the work greenhouse. I think most people either throw them out in the fall or bring them in and let them go dormant, so I'm not sure how practical of a suggestion this is, but it's at least technically feasible.
- Psychopsis cvv. (an orchid) (reader suggestion)
- Rosa cvv. (miniature roses) (reader suggestion)
- Streptocarpus cvv. (cape primrose) If grown under artificial light. (reader suggestion)
1 Abutilon only gets an honorable mention because they're not without their difficulties. My plants, all three varieties, are extremely messy, dropping leaves and spent flowers more or less constantly, which is annoying. They grow fast enough to replace the leaves they're dropping, but still. They're also inclined to flop over, to an obnoxious degree: I could probably fix this by keeping a fan on them, or staking the stems, but for whatever reason(s), I haven't done that. I also haven't pruned them the way I should -- I don't know what way one should prune Abutilons, but whatever it is, I know I haven't done it -- which means they're floppy, asymmetrical plants that would accumulate a pile of debris underneath them if I didn't move them to water every 7-14 days. (Instead, they leave a trail of debris from the basement, where I keep them, up the stairs and into the plant room.)
They also are extremely intolerant of drought, prone to whitefly, do poorly in heat, and need bright light and a lot of fertilizer in order to bloom, so they're not the easiest plants. For all their flaws, though, they're still one of the best fits for the category, because they really will bloom year-round.
Around here, not all establishments carry Abutilons, and those that do tend to have only the 'Bella' types, and they're only available in the spring at the start of the outdoor gardening season, but it might be different where you live. They can also be ordered on-line from a number of different places, if you're looking for something variegated or whatever.