The most common ones simmered along with curry are onions, carrots, and potatoes.
You could even add some sliced apples.
Root vegetables are frequently simmered with the curry, and you could consider using variants like sweet potato or squash, kabu, daikon, etc.
As a "topping", the sky is the limit; I've seen blanched okra, cooked renkon, roasted or simmered slices of kabocha squash, green beans, sweet potatoes, and cauliflower.
As non-vegetable toppings added after simmering, I've seen cheese, katsu-style tofu, tonkatsu or menchi katsu.
Edited to add: as Japanese curry is one of the easiest ways of encouraging our children to eat vegetables, my wife and I
now both add anything from blanched spinach (near the end of cooking) to (typically canned) tomatoes; the younger one loves it when there are plenty of green peas or even corn in the mix.
I like to add cauliflower.
cooking.stackexchange.com/users/2909/JasonTrue
In addition to the carrots and potatoes mentioned, I would recommend edamame.
You could also substitute lotus root for some or all of the potato.
And while you don't add this to the curry while cooking, you can top your curry with fukujinzuke for a nice cold crunchy contrast to the warm spicy curry.
cooking.stackexchange.com/users/11480/itoeats
Carrots and Potatoes at least.
Parsnip/Daikon Radish if you like those.
Frozen peas at the end are nice too.
cooking.stackexchange.com/users/13953/grumpasaurus
In addition to everything already mentioned, I sometimes see these used: Mushrooms Broccoli Tomato Eggplants
Stating the obvious: I would recommend against sweet potato if you don't want your dish to taste sweet.
cooking.stackexchange.com/users/2255/Nicolas Raoul
Traditionally, for home curries, the 3 common veggies are onions, potatoes and carrots.
You cut the onions in big 1-mouthful chunks (not finely diced - you want to be able to see them and eat them as a vegetable in the final dish), cube the potatoes to a biggish two-mouthful size, and cut the carrots into rough triangles - you do this by cutting the carrot as if you were thickly slicing it, but turning the carrot a quarter-turn before making each cut, resulting in even, chunky triangular bits.
Video demo of ran-giri here:)
The cooking time should be timed carefully after you put in the potatoes (last, just before adding liquid) as they can quickly dissolve into nothing if cooked too long.
Remember to serve with short/medium grained rice.
TIP: to make a richer tasting curry, add a small cube or two of dark chocolate after putting the roux in.
OTHER HOME VARIATIONS:
My personal preference is to use onion and a variety of mushrooms.
To make a milder, gentler curry for children, parents add a couple of tablespoons of honey, and a finely grated apple or two. '
Katsu-kare-' is placing a crunchy sliced hot schnitzel (like chicken or pork) on top of each serving of curry before serving...
it's delicious, but this is a lot more work.
Often curry is served with 'fukujin-zuke' red sweet veggie pickles on the side.
There are loads of interesting variations that people come up with, so it's fun to just experiment with what you have, too.
It's an easy, no-fuss family dish...
the equivalent of macaroni cheese, so using boxed roux isn't cheating; it's normal :)
You can make lots and eat it again the next day or two, as curry rice again, or on udon noodle soup (kare- udon).
We often came home after school and ate it with toast for afternoon tea lol.
You can't freeze it though, as the potatoes go very weird.
cooking.stackexchange.com/users/50922/lightawake
Small broccoli florets are nice as well (blanch them first, cut the stems short).
The crowns will hold a lot of the curry in, which is a plus!
cooking.stackexchange.com/users/43025/Nick
Depending on the day, I use a number of different methods and ingredients.
I make Japanese curry most days with S&B Medium Roux, but I also make it from scratch regularly as well which allows for the use of various curry powders.
We prefer either shrimp or chicken curry and the additions from what could be the standard Carrots, Potato and Onion include at times:
Shiitake mushrooms dehydrated and sauteed Frozen peas
Yukon vs russet potatoes Honey, Worchestershire sauce and soy sauce on occasion.
Dark chocolate sometimes Grated apple (frequently) Canned tomatoes I have spent a total of 15 years of my life back and forth in Japan from the time I was a small child.
We regularly had a variety of Japanese Curry, either in franchise restaurants, or made at home.
It can be extremely versatile and all up to your taste.
The best way to decide what to put in YOUR curry is to experiment.
Finally, it is obviously best served with JAPANESE type rice.
Short or medium grain, but short grain, while expensive, is really the best.
Best presentation is formed rice on one side of a large, shallow soup/salad bowl and the curry then spooned into the other side to ring the edges of the rice.
A good pinch of fukujinzuke along the edge or topping the rice makes the perfect dish!
Bon Appetit...and itadakimasu!
Added:
This question started me thinking about dinner and it gave me an itch, so I made exactly what we're talking about!
Included onions, Yukon Potatoes, carrots, celery, Shiitake mushrooms, frozen peas, grated apple, Worcestershire sauce, honey, and used S&B Medium Curry roux.
cooking.stackexchange.com/users/80502/SittingElf
I am preparing my first Japenese Curry (cheating incredibly by using the S&B Golden Curry Medium Hot box) and it suggests adding "vegetables" without actually indicating  what  veggies to add other than onions.  Assuming access to acceptable supermarkets (Asian and normal Australian) what vegetables (normal and/or exotic) make the best compliment to a curry like this?
What vegetables should I add to a japanese curry?
cooking.stackexchange.com
<vegetables><curry>
cooking.stackexchange.com/questions/28161
cooking.stackexchange.com/users/5923/Brian Ballsun-Stanton
The most commonly used are root vegetables like onions, carrots and potatoes, although you can supplement or substitute with variants like sweet potato or squash, kabu, daikon, edamame, lotus root etc. Curry is often served with 'Fukujin-zuke' red sweet veggie pickles on the side, as well as toppings including blanched broccoli florets or okra, dehydrated shiitake mushrooms, tomato, eggplant, peas and other ingredients. Root vegetables are often used. You can also serve with mushrooms or broccoli or many other vegetables.
Root vegetables like carrots and potatoes are very commonly used. Serve with mushrooms, broccoli, green beans or okra.