As I learned it my last game, it's apparently not common knowledge to ban Jax. Outplaying him and his whole team isn't enough; he's going to get gold and become unstoppable. But there has to be a way to stop him from killing literally anyone he encounters no matter how much health he has. I'm friggin' mad that I lost a perfectly good game because my team captain didn't ban Jax, but if it ever happens again, I want to know what can stop him. What counters him once he gets his inevitable core?

Yeah, I'm kinda heated right now, but I'm not just raging/crying. I really want to know what can stop him.

There's like one Jax-thread per page. Reading any of those should answer your questions so let's try something different here:
Which champs did your team consist of? What were the builds?

Best counter I've seen to him is Riven under a turret. your stun is on the cooldown as his jump then you use your dash and moving attack to run around the turret faster than he can attack you. Then when he runs away at low health you ult and kill him. I've done it a few times. Suprisingly the best way to kill him is to not attack him.

Cassiopeia and Ryze do work on him. Their sustained magic damage and ability to kite him can easily bring him down if you do it right.

CC abilities and Raduin's Omen/Frozen Hearth (armor + as reduction)

Once his core is done and he starts on defensive items there is very little you can do.

Ha, I feel like a jerk now. I looked, didn't see any on the front page, and decided to post, thinking, "Huh, this must be something everyone settled by now."

We had Teemo, Maokai, Skarner, Graves, and Wukong. Maokai was really shaky the whole game; he went bot and lost it a couple times. I was doing okay (Skarner), but I'd rushed Rageblade. I didn't even consider Jax coming back with a vengeance because he was terrible at the beginning. I don't know what the others were building; I think Teemo was on-hit.

He got to the point, pretty much automatically because you get gold for free, where from 10% health he could solo anyone. From at most 20% he could take me under my tower when I had full HP AND ulted him onto a mushroom. It was just so... anti-fun.

Pick Udyr and rape him.

Sword of the Divine.

yeah pretty much any potent kite character can kick Jax's ass due to the fact that his jump has such a long cooldown along with his stun relying on dodge. If you're melee facing him pick up thornmail and either Warden's mail or some magic resist. Most Jax's build AP so even though he's doing a **** ton of attacks it's the magic damage on the third attack and the empower ability that is giving him most of his life steal. Basically when I see an enemy Jax I build Thorn mail, Warden's mail, and Odins. This shuts him down hard and makes him have to go for someone else. Tanky builds in Dom are pretty potent, way more effective then damage builds.