How many companies or LLCs exist where at least one of the partners did not put up capital to receive equity in the LLC?

The prior answer was on the mark so I have nothing to add to that. Yes, you can award an equity stake based on equity stake. I'm no expert in corp. formation, but I've seen it done with a member-managed llc, where the members are awarded a percentage ownership of the company. But ownership and issues like deducting capital contributions invested into the LLC are different.

As the prior poster points out, sweat equity can be challenging to value against cash. I was in a scenario where an LLC awarded ownership percentage for an attorney's sweat equity. It was the right thing to do, but you'll have to stand behind those awards later. It may seem small you to now, but what looks like small parcels of ownership may look outsized to future investors.

For start-ups- most of the ones I see have at least 1 person contributing IP or something instead of a large cash payment.

Google "sweat equity"

I'm a partner in an LLC where I didn't invest capital, but I put in a significant portion of time in order to build infrastructure and software equity.

Of course that business didn't go anywhere so it was a waste, but I found that I valued my "capital investment" much more than the guys that just threw money in the hat (and the later wind down of the company pissed me off more than them since they just saw it as a bad bet whereas I pissed off my wife working long hours on a project that was stillborn).

Good luck squeezing the partners. Money brings out the worst in people and people never, ever want to share their pie. You're likely going to have to show how the product went from nothing when you got there to what it is today because of the sweat equity you poured into it... this is of course assuming that it was outside of your job description at the time. It can be argued that if you were "code monkey" and you were assigned to work on the project that you're due nothing since that was your job and anything they give you is just a gesture of good will. If you worked on it in your spare time, in a skunkworks type fashion, you'll be able to make a better case.

ETA: I've worked at a couple startups where a founder was not a capital partner, but was a sweat equity partner due to past work, but I'm not sure I can disclose them. They do exist.