I updated this guide on 12/31/11. I stopped playing Jax for a long time, came back, and made some observations that caused me to change the build and comments a bit. Unfortunately, dodge will be removed from the game soonish and I'll have to redo the build, most likely. Ah well.

Anyway, here are some observations about Jax from personal experience, chat, and other guides:

Life doesn't begin until level 6

Debatable, I guess. Since some people can be aggressive with him successfully at level 1, that's fine, but I don't seem to do well that way.

Jax does a lot more damage with the passive on R, and activating it lets you engage with mage carries. Even if you manage to harass well with Counter Strike and Leap Strike, it isn't particularly safe or particularly devastating before then.

I think with at least six levels and 30% bonus AS and Ninja Tabi, he's in good shape, though. I throw in a Philosopher's Stone because I prefer to throw in Hextech Gunblade late, when his DPS is high enough for the dual-vamp to really shine; you can't dual-vamp if you're hitting them with a wet noodle. Also, Gunblade has been nerfed lately, but the active and the dual-vamp make it good enough as a late item.

Jax needs items

If you don't have attack speed and Ninja Tabi, you should be mostly worried about last-hitting minions. Sometimes you can pick up kills, but getting greedy early will probably mess you up more often than not unless you know what you're doing and the odds are in your favor, or you know what your lane mate is capable of, whatever.

Exhaust is sort of mandatory

Jax needs to land multiples of three hits to do any damage worth mentioning. So Exhaust is pretty important. Ignite is nice if you're worried about healing, but that's negotiable. Ghost is probably very solid.

Minions are good for building stacks

The ideal situation is that you can farm away and then Leap Strike into a champ with your 10 stacks of Relentless Assault ready to go. They should be reluctant to engage you if they see you auto-attacking away, though, and they won't have to wait long for the stacks to fade away. Jax is particularly well-suited to auto-attacking super minions, and he does solo Dragon acceptably well at 11 and only gets crazy fast at it from there. ( Guinsoo's Rageblade and two points in Relentless Assault is probably the soonest that it's worth the risk of soloing it).

The red buff is your friend

Jax is especially well-suited to getting the red buff, especially for the slow effect. The damage over time helps too, of course. The blue buff is almost surely better used on someone else, with your cooldowns being about as short as they need to be to begin with. Some people suggest Rylai's Crystal Scepter and it's a good choice, but I prefer keeping an eye out for the red buff, since Jax can take out the lizard early (probably at 6, haven't tried yet) and quickly.

Sheen is nice at high levels

Jax has *two* skills with low cooldowns, once they're leveled up some. So between Relentless Assault, Empored Strike, Leap Strike, and Sheen, you should be doing some weird cocktail of regular attacks and tiny bursts, with the bursts happening so often that it's like the popping of popcorn over the sizzle of auto-attack butter. mmmmm, triple kill, hits the spot.

1v1 at mid-to-high levels with a squishy is ideal

I'm just saying that if that Miss Fortune tries to 1v1 you, then she has no idea what Jax is and you should enjoy the free kill. You should probably also give ganking a try if you have any enemy champs with no real escape mechanisms.

I redid the item order a little, and I promoted Malady to a main item instead of a situational. The main reason is that Relentless Assault can be activated for adequate bonus MR, and Malady has a ton of synergy with Jax.

If you get something else on Jax, you're a troll. Boots and some Health Potion should give you a decent amount of lane time. I don't seem to have mana trouble, since I'm not a very aggressive Jax player early on.

philosopher's stone The mana isn't really necessary, but it's nice to not even have to think about mana. The GP10 makes this a good throwaway choice, and the HP5 makes a big difference and lets you delay Hextech Gunblade. Hit-and-run Leap Strike harassment goes better when you're healing all the time.

or Attack speed is mandatory. There's almost always enough magic damage in the game to justify picking up Wit's End, but Malady will really help your DPS snowball to dangerous heights. Taken before Rageblade because you need to be quick enough to build up the stacks in the first place.

Ubiquitous for a reason. Cost-effective, adds to the speed and damage of your R.

Also ubiquitous for a reason. The AD and AP work with your auto-attack and they work with your passive. Don't forget to use the active - it helps you get more auto-attacks in, building up your R and your Rageblade. I do NOT like getting this early, though! It's too pricey and can't really help much unless you have attack speed anyway, and the life steal/spell vamp synergizes better with a solid, pre-existing DPS foundation. It used to be worth getting this item earlier when it had a lot more AD, but now it's still worth it on Jax late-game and... not that many other champs that I know of.

Everything here helps just a little all around. Sheen effect makes you a little punchier. Phage effect should proc often enough if you use Hextech Gunblade and/or Exhaust and Counter Strike for some free hits to get started. Zeal helps you spend even less time running between hits. They have to CC you or else.

You probably won't get this far, but if you do, you'll already have a decent health pool, be doing tons of damage, and hopefully out-vamping them most of the time. A bunch of resists and the ability to pop back to life again with 750 health should just about break the game.

Situational

I wouldn't mind Eleisa's Miracle, and sometimes I get it if I'm lazy or it's almost over and I have exactly enough gold. But the attack speed to get the ball rolling again is probably more valuable, and the CC reduction seems like a good idea for not losing stacks too much. Once you have Gunblade and Trinity, your mana is good and you're recovering health in combat, so you're free to sell philosopher's stone at will. Cloak and Dagger is a bit of a throwaway item, but I hate getting CCed a lot.

If CC isn't an issue, this is more useful than Cloak and Dagger

If they stack health, for example, especially if you already have Malady.

I would like to experiment with this as a sixth item someday. I think it might be fun but not the best choice. Haven't tested it.

This is kind of tempting, but not really that tempting, the more I experiment with other options. Everything in my main build seems to contribute more.

This could be fun if the enemy team stacks too much AD. Might be sort of troll, but I had an enemy team with no AP once, and it seemed to help.

I might try to work this in, but there's no room for it right now. CDR can let you proc Sheen like crazy while landing more stuns, leap strikes, etc. Armor pen is nice, and the active lets you start your stack-building as if you already had stacks. Move speed keeps them from getting away... it's just crazy good. But it's a bit too AD-oriented for what I have here at the moment, and the AS would be overkill.

I want to test this. I should've thought of it sooner - activating this should be perfect for getting more auto-attacks in and stretching the time between Leap Strikes better in your favor. Just Warden's Mail alone should be brilliant for dueling against AD champs, I would imagine.

Heart of Gold might not be a good addition to Philosopher's Stone unless you're really suffering in lane and need the help both staying alive and making gold to make up for getting denied hard; Jax snowballs so hard that a second GP10 item probably won't be worth it, generally, and you'll get health from Relentless Assault anyway.

Leap Strike > Empower > Counter Strike

Leap Strike first because it has much more utility and it reduces cooldown. Empower next because it also has reduced cooldown. Counter Strike at 1 is fine with Ninja Tabi.

Of course, Relentless Assault is insane and needs to be leveled whenever you can, i.e., at 6, 11, and 16.

Cleanse is probably good. Leap Strike + Counter Strike + Flash is probably good for emergency aid to allies when it's dangerous to stick around, or you can Flash as a second Leap Strike to keep your stacks up or when you're not quite in Leap Strike range. Rally together with Exhaust should be pretty deadly, since you use AP and AD with your auto-attacks thanks to the passive on R. Ghost is probably really solid for keeping the ball rolling when they want to run.

Nothing else really stands out at the moment.

21/0/9 for dual-pen, AP per level, enchanced Exhaust, AS, % damage increase, move speed, and red buff duration. New masteries are more hybrid-friendly than they used to be, mmmmmmm.

Runes: New dual-pen runes have to be fantastic. MR/level glyphs make up for skipping Wit's End somewhat. Dodge runes for sure, if you still have them. Otherwise, move quints and attack speed marks can do, I suppose.

Unless my lane mate is especially dominant, I like to turret hug early, and Leap Strike and Empower to last-hit minions early on. Sometimes they overextend and I can finish them off with Exhaust, Ignite, and QWE, but I don't spam skills, generally. Saving your spells for when you can get a kill helps a lot.

Around the time I complete Guinsoo's Rageblade, my playstyle becomes more Rambo-ish; it helps to have Vampiric Scepter by then, too. It's hard not to play sort of Rambo since you have to build Relentless Assault stacks. A lot of the time, I tend to get 1.5 kills and then die, leading to final scores like 10-7-6 or whatever, but with a little more discretion this could probably be improved.

Pros:

Strong late game
* Somewhat sturdy, but plays almost like a carry
* Can shred people if there's enough taunt/stun on the team to make it happen
* One of the best 1v1 champs in the game when decently fed

Cons:

Weak early game
* Needs 2-3 items badly
* Especially vulnerable to CC
* Can be countered: CC, or run away; anything but sitting there and taking it in the face limits his DPS (it's harder for them to run away, though, later on when you have more sources of slow from items). Also, the obvious solution to 1v1 gods like him is to focus them when the time is right.