I didn’t even realize mayo and mustard were an either/or option! I wouldn’t make a sandwich (hamburger included) without adding both. I’m sort of horrified at the thought of having to choose.
What about avocados? They’re still in season and taste wonderful on burgers…
Sliced jalapenos.
Not mentioned so far- tartar sauce, horseradish sauce, 1000 Island dressing. The 1000 Isle with a bit of horseradish is nigh equivalent to Big Mac special sauce.
Not just raw onions, but also grilled.
Beetroot
Pineapple
Fried egg
Don’t forget the beetroot and pineapple! Also mushrooms! And an egg if you can fit it on. Lots of salad. OMG I want one now.
Yuuum
Damn, I wanted to offer that. If this was a contest,you win hands down.
I also vote for fried eggs.
Ahhh yes, the bun. McDonald’s has better buns than Burger King. Wendy’s tries too hard. Cheeburger Cheeburger has the best. They’re ike Wendy’s but fresh and substantial.
Steak Sauce
Worcestershire Sauce
Sliced bell peppers
Fried egg
Another vote for the pineapple, bacon, beetroot and fried egg!
If you have a Fuddrucker’s anywhere near you, buy their buns. They’re fresh-baked and nice and chewy and don’t fall apart into little fluffy crumbs.
I’ve long been of the opinion that any hamburger that doesn’t include these ingredients promotes Communism. 
Both my items have been said.
Bacon
Hamburger buns - they do make varieties of them too, so don’t feel obligated to only buy plain white buns.
Most people can make a hamburger they like from what you already have.
You do have black pepper and salt right?
This stuff is all slightly more obscure, but still good on a burger:
Blue cheese crumbles
Sauteed mushrooms
Montreal Steak Seasoning for the rub
A-1
And if you don’t want to put bacon ON the burgers, try putting it IN them.
This is why Australia will never be a world power.
I think you’ve covered all the standard bases. I’d like some pickled hot peppers, as well, but that’s not a “standard” topping. Ditto on the fried eggs. Awesome on burgers, but it’ll complicate the whole cooking process (I’m assuming you’re grilling out, and you probably don’t want to worry about making fried eggs along with the burgers, no matter how easy it is). That, and I think the average American would consider eggs a weird topping for a burger.
Jalapenos and horseradish for the adventurous souls.
I like it hot, too. I’ll put giardiniera on anything resembling a sandwich. Sometimes, though, it can get to be too hard to get to stay on the sandwich, especially burgers. I usually opt instead for a couple swirls of sriracha after ketchup application.
Mayo for Life.
I like Thousand Island on mine, unless it’s primo beef, in which case A1.
But OP, try some Worschester sauce on the outside of the burgers before they are cooked.