What kind of bakery items do not get horribly affected by electricity going off when the item is still in oven?

On a different tack to those above you can stabilize your oven temperature by using heat absorbing materials such as brick and stone, which will absorb heat when the oven is on, then release it when the power goes out. It does mean for longer pre-heating times as it takes lots of energy to get the stone or brick up to temperature, however it could save your cakes. Just make sure to use fireplace brick and not regular, as regular can crack. Better yet, get a piece of scrap granite or marble, that way you can use it as a bread stone as well.

Yeast-leavened breads (no matter whether they're white, whole wheat, or multigrain) are going to be your best bet. They have longer baking times, so a power outage will represent a smaller fraction of it, and the yeast keeps on working over time. So if the power goes out when the outside structure is set but it's not baked through yet, the inside won't just collapse. You'll still be a lot better off with a 15 minute power outage than an hour, but I think you could probably manage through longer ones too. The tricky thing will be telling when it's done - you obviously won't be able to trust a time from a recipe anymore, so you'll have to rely on testing. Depending on the specifics, you may also end up having to cover the top with foil or even reduce the temperature a bit to finish baking, to avoid over-browning it.

Quick breads - things like muffins and pound-cakes, soda-leavened with a dense crumb - will also fare okay. For example, I've had the power go out halfway through a 30-40 minute baking time on muffins, and just let them sit in the still-hot oven for the rest of the time plus an extra 5-10 minutes, and they came out okay. Not perfect, but I certainly wouldn't complain. If on the other hand you have a long power outage, especially early on in baking, you might be out of luck. If the leavening is spent before it cooks enough to set up some structure, it'll collapse on you.

If the outages are really short, on the order of 5 minutes, I wouldn't worry about much of anything at all as long as the baking time is longer. The oven already cycles power on and off to maintain temperature; with an outage that short it won't have time to drop before it gets a change to recover.

It also helps if your oven is good - better ovens hold their temperature better, releasing less heat to the surroundings, so it'll take longer when the power goes out for them to drop to a too-low temperature.

Finally, I would encourage you to experiment a bit. If you don't mind if 1 in 4 times something comes out a bit messed up but still quite edible, just go for it. And having things come out less than perfect will help give you an idea how much you can get away with, so you don't spend all your time avoiding things that you actually could have made!

Going the opposite direction from that recommended by Jefromi in his answer should also be very effective: bake items with shorter bake times, like cookies (or, as the British say, biscuits) or mini-muffins.

These items, with their short bake times, are less likely to be interrupted by a power outage, and if an outage does occur, the assuming the oven was properly pre-heated, the carry-over heat should allow the individual tray to finish baking, even if it takes a minute or two longer. As mentioned, Jefromi ovens cycle on and off anyway during normal operation.

Most of these items should be fairly tolerant of holding the dough for the next tray or batch until power comes back, within reason. They actually tend to benefit from longer hold times due to the additional hydration of the flour, and the mellowing of some flavors (especially cocoa based items).

Note: you have implied though never specifically said in previous questions that you have no refrigeration. Cake and cookie doughs often contain eggs, which are a fairly perishable item. You need to be cognizant of how long you are holding raw dough or batter containing uncooked eggs. An hour or two at moderate temperature is probably fine, assuming they then get cooked properly to an internal temperature above 160 F - 180 F (which should be the case in all cookies and baked goods).

Ironically, meringues, which are essentially nothing but egg whites and sugar may also fair very well for you, as they are very tolerant (and actually benefit from) a long, slow drying. Even if the power goes out, leaving them in the cooling oven should work. However, you don't want to hold uncooked meringue for too long.

Anything that will fit into a steamer on your gas burner, wrapped if necessary.