"Looks like you can't go infinite before turn four, so this is probably just won by the most aggressive strategy" - a quote from myself. I was wrong. You can totally go infinite on turn 3 on the play.
Turn 3 on the Play, Main Phase 1: Infinite Damage
Turn 1: Yavimaya Coast, Honored Hierarch (5 cards)
Turn 2: Wooded Foothills > Mountain, Atarka’s Command, dealing 3 and putting a Windswept Heath into play (3 cards, 2 gy)
Windswept Heath > Plains, Defiant Strike on Hierarch, draw 1, attack with Hierarch becoming renowned (3 cards, 4 gy)
Turn 3: Polluted Delta > Island (3 cards, 5 gy)
Tap Island, Plains, Mountain for Jeskai Ascendancy (2 cards, 5 gy)
Tap HH for R, play Magmatic Insight, discarding Swamp, draw two, untap HH, loot a card (2 cards, 8 gy)
Tap HH for U, play Treasure Cruise delving 7, untap HH, loot a card (4 cards, 3 gy)
Tap HH for R, play Magmatic Insight, discarding Swamp, draw two, untap HH, loot a card (4 cards, 6 gy)
Tap HH for R, play Spidersilk Net, untap HH, loot a card (R, 3 cards, 7 gy)
Tap HH for W, tap Yavimiaya Coast for U, play Jeskai Ascendency, untap HH, loot a card (2 cards, 8 gy)
Tap HH for U, play Treasure Cruise delving 7, untap/tap HH for UW, loot twice (UW, 4 cards, 4 gy)
Play Spidersilk Net, untap/tap HH for UR, loot twice (UUWR, 3 cards, 6 gy)
Play Jeskai Ascendency, untap/tap HH for WR, loot once (UWR, 2 cards, 7 gy)
Play Jeskai Ascendency, untap/tap HH for UUW (UUW, 1 card, 7 gy)
Play Treasure Cruise delving 7, untap/tap HH for UUUW (UUUUUWW, 3 cards)
Play Soulfire Grand Master (UUUW, 2 cards)
Play Defiant Strike on HH, draw one, untap/tap HH for UUUU (UUUUUUU, 2 cards)
Play Disperse targeting Spidersilk Net for UU, trigger Soulfire Grand Master for UUUU to return Disperse to your hand, untap/tap HH for UUUU (UUUUU, 1 card, Spidersilk Net, Disperse)
Play Spidersilk Net, untap/tap HH for UURR (UUUUUUURR, 1 card, Disperse)
Repeat the previous two steps an infinite number of times, gaining RR each time until you have infinite R (Infinite R, 1 card, Disperse)
Play Rolling Thunder for X = infinite, targeting your opponent for all of it, dealing infinite damage and gaining infinite life as well
But maybe Jeskai Ascendancy and Treasure Cruise are too easy; after all, both have seen play in Legacy (and even Vintage). Lets go infinite without using a single non-land card that's seen play outside of Standard.
Turn 3 on the Play, Main Phase 1: Another Infinite Damage Combo
Turn 1: Forest, Obscuring Aether (5 cards)
Turn 2: Forest, Obscuring Aether x2, Rattleclaw Mystic x3 face down (0 cards)
Turn 3: Unmorph Rattleclaw Mystic #1 for GG, add RGU, tap for U. Play Secret Plans for GU. (RU, 0 cards)
Unmorph Rattleclaw Mystic #2 for RU, add RGU, draw a card, tap for U (RGUU, 1 card)
Unmorph Rattleclaw Mystic #3 for RG, add RGU, draw a card, tap for U (RGUUUU, 2 cards)
Play Rattleclaw Mystic #4 face down, unmorph it for RU, add RGU, draw a card (RGGUUUU, 2 cards)
Play Temur Ascendancy for RGU, tap Mystic #4 for R (RGUUU, 1 card)
Play Clutch of Currents for U, returning Mystic #1, replay it face down, unmorph Mystic #1 for RG, add RGU, draw a card, tap for U (RGUUUU, 1 card)
Repeat the previous step three more times (RGUUUUUUU, 1 card)
Play Sidisi's Faithful, exploiting itself for U, returning Mystic #1, replay it face down, unmorph Mystic #1 for RG, add RGU, draw a card, tap for U (RGUUUUUUUU, 1 card)
Repeat the previous step three more times (RGUUUUUUUUUUU, 1 card)
Play Learn from the Past for UUUU, targeting yourself, shuffling in everything and drawing a card (RGUUUUUUU, 1 card)
Repeat the above cycles with Clutch x4, Sidisi's Faithful x4, gaining U per card and drawing one off each (RGUUUUUUUUUUUUUUU, 1 card)
Play Learn from the Past for UUUU, targeting yourself, shuffling in everything (including the other Learn from the Past) and drawing a card (RGUUUUUUUUUUU, 1 card)
Repeat the above two steps an infinite number of times, gaining infinite U (RG, infinite U, 1 card)
Play Crater's Claws for X = infinite, targeting your opponent, dealing them infinite damage
Previous Solution
39 damage Turn 3 on the play.
Turn 1: Wooded Foothills, crack it for a Forest, Scythe Leopard. (5 cards, 1 in yard, 0 damage)
Turn 2: Wooded Foothills, crack it for a Mountain, Atarka's Command dealing 3 and putting Wooded Foothills in play, crack it for Canopy Vista, Defiant Strike on Scythe Leopard. Attack with Scythe Leopard for 6 damage. (3 cards, 5 in yard, 9 damage this turn)
Turn 3: Wooded Foothills, crack it for a Cinder Glade, Animist's Awakening for X = 3, revealing Windswept Heath x3, crack for Cinder Glade x3, Become Immense on Scythe Leopard, delving 5, Temur Battle Rage on Scythe Leopard, giving double strike. Attack with Scythe Leopard for 15+15 damage. (0 cards, 7 in yard, 30 damage this turn)