I currently have an 11-28 cassette on a compact Shimano r7000 105 (50-34). I'm wanting to move to a cassette with a larger rear cog. What can I upsize to without requiring a new chain?

Nothing. The bigger cogs will mandate a longer chain.

You need the rear derailleur with the medium cage to handle a 32 or 34-tooth cassette, fyi. Shimano claims that it maxes out at 34t.

Yeah, assuming that the chain was properly sized in the first place, you not going to be able to go bigger without a new chain. There is a chance that it wasn't properly sized, though; shift to your big/big and look at your RD - it should be stretched forward with some, but not a lot, of bend at the elbow. If there's a lot of bend, you might have a couple extra chain links to play with.

How old is your current chain? Unless it is almost new it is never a good idea to install a new cassette without changing the chain as well

There is nothing improper about sizing a chain using the longest chain method of small+small w/o slack as long as you are within the capacity limits of the RD

That's... not what he said.

We're in Road. We're weenies. It's improper to weenies.

This is also true.

He said that assuming it is properly sized, that you wouldn't be able to go larger without a new chain, with an r7000 SS that has a capacity for a 11-30, a chain sized using small+small would work for anything from an 11-25 to an 11-30 without needing the length to be changed. Since the OP is using an 11-28, if the chain was sized using this method then they could go to 11-30 without changing their chain

I know campy makes gravel equipment now but they are still primarily road. Is their chain sizing method not appropriate?

Campy? What's that?

The drivetrain is all fairly new. It has ~1500 miles on it. It's all the factory stuff that came on my 2018 Specialized Allez Sprint.

So the takeaway here is that I can go 11-30 without changing chain or rear mech, but if I want to go 32 or 34, then it would require a new chain and mech?

The takeaway is that we don't know how the chain was sized originally because there are a few different ways to correctly size a chain. Since it is a KMC chain you can test it easily by installing the new cassette and wrapping the chain around the big front cog and big rear cog and see how much excess chain there is for your desired cassette. The rear mech is generally conservative so an 11-32 should work fine on a short cage mech and possibly even the 11-34.

That's not quite what was said. Given that we don't know how your current chain length was measured, your drivetrain might be at its limit with your current 28 sprocket. A 30 sprocket will be within the chain wrap capacity of your current RD, so you know that you're not going to need a new RD if your switch cassettes, but again - this is a separate issue from chain length, which is the one individual variable about which we know nothing. You will not need a new RD, and you may not need new chain. Easiest thing to do is put your drivetrain on large/large and confirm that your RD is not at its limit. If it still has a bit more forward travel, you likely will get away with your current chain

Maybe, but not likely. The most common sizing method for road, and what Shimano recommends, is big/big + 2; this would leave your chain too short for the additional teeth.

Most likely yes on the new chain, not necessarily on the new RD - I haven't tried it, but many people seem to get on just fine when exceeding Shimano's largest cog spec by two teeth (check to see what the official spec is). Alternatively, you might be able to use something like a Wolftooth Road Link as long as the RD can handle the wrap.

Regarding chain length sometimes you get lucky. I wanted a more robust spare wheelset and run easier gearing on my Dura-Ace Di2 Roubaix and was able to handle an 11-32 cassette without changing the chain even though the bike was supplied with 11-30. I found this online tool helps with approximating chain length.
Javascript Bicycle Chain Length Calculator

It cost me $400 to upgrade from a short cage 11-28 drivetrain to an Ultegra 8000 long cage RD, 11-34 cassette and chain installed by my LBS.

IMHO, worth the extra $ to have a finely tuned drivetrain.

Thanks for all the help, guys. My next question: is it even worth it to try such an incremental change, or should try a 32 or 34 first? I feel like I'm exerting too much energy powering up any grade >6%. I see guys on 34-36 having an easier time at it. I want to change my cassette not because I want to keep up with the Jones', but because my legs feel exhausted by the time I plateau.

You answered your own question.

I have been able to complete several long ascent bucket list rides because of the 11-34 cassette offering 27, 30 and 34 cogs with my 34 chain ring. Sometimes you need the climbing cogs to 'rest' the legs. I was really fatigued always grinding in the 25, 28 cogs.

I guess I did ...looks like I'll need a new chain, mech and cassette...

It's called a "long cage" mech, right?

I wore out a chain in about 1500 miles a few years ago. Chains are cheap enough that it's a no-brainer to get a new one when you buy a different cassette. You could always put the old one in a baggie and save it in case you ever want to put the old cassette back on.

GS, but I'd try it with the mech on there already.

This is how I understand it.

I have 9000 and ride 11-32 cassette on it. I ride a Trek and they seem to have long hangers. Giants do also.

I ride a 2018 Trek Domane AL3. It came with a 11-28 rear cassette, with Shimano Sora 9 speed group set. I installed a 11-32, 9 speed cassette on the rear, and have been riding for a year now without a new chain. Seems to work fine for me. And I do occasionally go big/big. Never had an issue.

Good to hear. I've been told to avoid cross-chaining, so I avoid going big-big, small-small and the next up/down combination. This essentially makes the bike a 16-speed instead of 22; go figure