Hello, welcome to Maxed Jax. You might look at the item build and freak out because it is so expensive. But in all truthfulness, you only need about 250 CS, and a fair amount of kills and assists to finish the build. Once you have 3 or 4 items of it, you will be a 1-man army. This build truly does max out Jax's potential 100%. Jax synergises with the most expensive item set better than any other character.

For the runes you will want critical strike chance marks, they up your damage output by a ton. Dodge seals for obvious reasons. AP per level glyphs since they give almost 30 AP at level 18. The quintessences, one might argue that the dodge ones are better than more critical strike chance... But I get by just fine with 23ish percent dodge chance. If you want more dodge chance, by all means use the dodge quints, but it's really not necessary and if you really think you need more dodge chance, you're probably doing something wrong.

Jax puts out a good amount of physical and magical damage. So 27/0/3 is the way to go. Slap the trinity force onto the mana Regen you get from Perseverance, and you will feel like you're using the ancient golem buff.

I did quite a bit of calculations on the items. The item build I created is just so monstrous and powerful. Here, have a look.

These are items I considered adding but decided not to have them:
Youmuu's Ghostblade
Phantom Dancer

Actually, PD and YG were trying to take over Trinity Force's spot! I felt that Trinity Force had too much utility to pass up and not only that, PD and YG gave just a little bit too much attack speed. The build with Trinity Force, already has a attack speed of 2.4 with maxed out Guinsoo's Rageblade stacks, and maxed out Relentless Assault stacks. (plus the green potion)
Another reason is because Trinity Force's passive adds 317 (physical) damage for an attack. CD for this is like 2 or 3 seconds.


EDIT: You can really just disregard all of that ^ The core items of the build are as follows:

Hextech Gunblade
Trinity Force
Infinity Edge
Rabadon's Deathcap

Situational Items:

Against a balanced team, get Mercury Treads! Against a team that has 3 AD champions, get Ninja Tabi!

As Jax, you have to have a good defensive item that counters the enemy team. If they are mostly AD, get Thornmail. If they are a balanced team, get GA. If they are mostly AP characters, get Abyssal Scepter.

Do the math, late game you'll be a semi-tank spamming about 500 damage abilities with a max attack speed of 2.3 or so. You'll be going on crit streaks critting for at least 800, and every other hit will deal around 500 more damage plus with every ability used your next hit deals around 300 more damage. Scary....

Also, just some clarification on why I scrapped Rageblade:
It's a bad item. It does not give Jax the burst he needs. 30 some AP and 40 something AD is bad before you gain any stacks is just bad. And once you do have ALL the stacks, having a SECOND gunblade would be better still! Rageblade is cheap, but it's a waste of time to get and sell later. Focus on Trinity Force items early game that is the best way to go. Having mana issues? get the sapphire crystal. Getting harassed too much? Ruby Crystal. ECT. ECT.

Jax's harrass comes from his W + Q combo, so it's a good idea to focus on these first! Later in the game, after the laning phase, you should definitely focus E for effective jungling.
The reason I focus Q more than W is because Q simply does more damage.

It's not just about Ignite and Exhaust. It's about the fact that they give me 10 AP, 10 ArP, and 10 magic penetration. Well the 10 ArP and 10 magic pen. are only for the duration of exhaust but still. Jax is monstrous, slap the two best offensive summoner spells on him and he's unkillable. Ghost, Teleport, and Flash are good. As many people say: Summoner Spells is a matter of preference, I say it again. But really, you should not be taking any of spells I haven't mentioned ^
Except maybe smite, never tried jungling with this build however I know that Jax is a pretty good jungler.

For ranked play, your early game will be even harder, but as long as you get through it, you'll be fine.

Pros:
*Maxes Jax's damage output.
*impossible to lose a 1v1
*AoE Stun
Cons:
*hard early game
*Jax doesn't bring a whole lot to his team other than his amazing dueling advantage

I start jungling at level 6+ AND I always make sure I have some sort of life steal or spell vamp before I jungle.

Jax is capable of GREAT things! Example: Caitlyn was targeting my team mate who had about 100 HP left. I used q on my team mate just in time, and the bullet hit me instead. My team mate was so happy! Also in another scenario, the team fight was about to begin... Enemy twitch was nowhere to be seen. Shen came around the corner (takes place in jungle) and attacked us! My team mates were stupid enough to try to kill him when twitch popped up at the other side of a wall and sprayed and prayed and oh god I was getting nervous. But never fear, Jax is here... I leap striked over the wall at twitch and killed him rather instantly.

Impossible to lose a 1v1 late game. Beast Mode = Activated.

I jump into a big pile of creeps, E activates somewhat instantly. Try to farm when there isn't a lot of tension going around. If you are needed, you should probably sacrifice farming temporarily.

If you like being the guy everyone is afraid of 1v1'ing or defending against alone, use this build, or if you want just be a good Jax player. But this build isn't going to give you some kind of magical ability to dominate. You need to be good at the game to.

Don't jungle when you should be pushing or the enemies are pushing. Don't initiate the fight, let the tank do that. Be weary of skill shots. Using Q on an ally can save you from Blitz's grab, Ashe's arrow ECT. Make sure you get your early and early-mid game farm, you will need it.