BatsAt the minimum, you should be bringing your main competition bat, and a backup bat. Preferably the backup bat should be the exact same type of blade and rubbers as your main competition bat. You don't want to be trying to adjust to a different blade and/or rubbers in the middle of an important match, if you are unlucky enough to damage your main racket.
I take things a step further, and I take turns using both of my bats, so that the rubbers are always roughly the same in terms of wear. I am also fortunate that I don't speed glue my rubbers, or else I would also be speed gluing my backup bat when I play competitions. Yes, it can be a waste of glue, but once again I would hate to damage my main bat and have to use a backup bat that isn't speed glued.
ShoesI bring two pairs of table tennis shoes in my bag (again, the same brand of shoes). I'm less likely to damage a shoe than a racket, so I tend to use one pair of shoes for training purposes, and save the other pair for competitions. When my training shoes get a little worn out, I throw them away, make my competition shoes my training shoes, and buy a new pair of competition shoes. Putting on my best shoes for a competition is part of my preparation ritual - when the new shoes go on, it's time to get serious out there!
Racket CleanerThe best racket in the world won't help you if it's too dirty to grip the ball properly. I always carry a rubber cleaner with me so that I can make sure my rubber is nice and grippy. My own cleaner is a carpet cleaning sponge in a convenient plastic case - I can keep it damp and ready to go without getting water on my equipment or stinking out my bag. You can also use a plain dishwashing sponge in a ziplock bag.
I personally don't bother these days with the racket cleaning foams, sprays etc. I used to use these in the past, but these days I find simple water and a good sponge do the job just fine.
Spare GlassesAs a person who wears glasses, I always bring along a spare pair just in case. I would find it pretty difficult to play without them, so I don't risk it. Contact lens wearers should probably bring a spare pair of contacts (or a backup pair of glasses) for the same reason. You might also want to bring a cloth to clean your glasses with (although I tend to clean them with my spare shirt, which is nice soft cloth anyway), or spare lens cleaning solution.
Change of ClothesLike most players, I usually wear my outfit and a tracksuit to the competition. I also like to bring along at least one spare pair of socks, and usually a spare shirt (or two), shorts, and even a change of underwear. I don't always use every item, it depends on how much I'm sweating, and how many events I am in. But I usually like to change into a fresh shirt around semifinal time, when the matches are getting serious, and I'll generally change my socks at this time as well. On a really hot day I'll change my whole outfit, so that I feel fresh, clean, and mentally ready to get going again.
TowelI always have at least one big bathroom sized towel in my bag. It's one of those things I always look for in my bag before leaving the house. There's nothing like having to wipe your face on your shirtsleeve all day to help you remember your towel in the future!
GlueI carry a spare tube of ordinary rubber glue, just for minor touchups where the rubber may have come away from the blade a little around the edge, or if I clip the edge of the bat on the table and peel a little bit of the rubber loose. Back when I was speed gluing, I always carried a nearly full tin of speed glue in my bag, as an emergency supply if I ever needed to reglue.
Bat PressWhen I was speed gluing, a bat press was part of my equipment that always came along for the ride, just in case I needed to reglue (especially during a long day, where I would reglue after 4 hours or so).
Nowadays, I don't often bother, since my bat press is heavy and I don't need to reglue my rubbers anymore.