1 is: Spirit of the Elder Lizard, IE, BotRK, IBG, GA, Boots of Lucidity. Another i've seen is more all out offense and it's: Spirit of the Elder Lizard, IE, LW, IBG, BotRK, Boots of Lucidity
Some versions have BT or Statikk Shiv. It's basically Q spam all day with red lizard buff for lane dominance.
So what happens when you get to lategame? No crits, no damage?
Of course you don't go this build for lategame? Why do people even ask questions like this. Someone would go this route to do enough damage early/mid to offset or delay a late game before their team is far enough head/winning etc. --- LoL IGN: Greedy
If this is a thing then Ez isn't used for pure damage. They use it for a poke comp to drop people's HP down before a teamfight, then have AP carries and bruisers ready to get in there and dunk people to hell while Ez provides support fire from behind. --- ~Victory needs no explanation; defeat allows none.~
The reason of the strange items is because it's focus on IBG and CDR. At the early level, Spirit of the Elder Lizard will provide him all the health and mana regen necessary for the laning phase while also giving him damage and CDR.
By the time you complete the 3 main items, Spirit of the Elder Lizard, Iceborn Gauntlet and Boots of Lucidity, you will have 39% CDR reduction (additional 4% from mastery). This will put Ezreal's Mystic Shots at 2.4 seconds (1.4 if it hits) and Arcane shift at less than 7 seconds(it's also receiving 1 second CDR from every Mystic Shot landed). He can basically become the most slippery, un-focus-able champion in teamfight. He will poke and Arcane Shift away easily. If you have Lux or any other hard to focus Champion as APC, the enemy will confuse on who you will focus in teamfight. And Ezreal's IBG can slow the entire enemy frontline. The items are allowing him to steal dragon and baron with his R + Q from outside the wall.
The Spirit of Elder Lizard will add extra 25% damage to his skills. And the burns, at max level, you can see 44 damage popping up from the enemy that hit by Mystic Shot if you have red buff. And you can spam it.
After you complete the 3 items, you can get BT/BotRK to have a better sustain. You can get Vamp Scepter before completing IBG if necessary. I personally goes for BT since it's easier to finish people off, but BotRK will allow you to kite someone until he stress out and give up. The next items will be IE and LW.
If the game extended, and kiting become an unnecessary since your teammate will just go in and die(without even trying to run after getting low) in teamfights, you might want to sell some items for a standard ADC items. Maybe you can give it a try and post the experience.