1 turn, 8 cards from hand + 2 from library
Forest, Oviya Pashiri, Sage Lifecrafter; 3×Mox Amber, tapping each for {G}; Burning-Tree Emissary, after which one has {R}{G}{G}; Chaos Warp targeting Mox Amber, putting Omniscience onto the battlefield.
Now cast Acererak the Archlich repeatedly, going through Lost Mine of Phandelver and getting a Treasure each time. This is not quite infinite yet because that dungeon also makes one draw a card each time, so add Ormos, Archive Keeper to stop that from causing a loss and make it truly infinite.
1 turn, 18 cards, relying only on drawn cards
Phyrexian Tower, Ornithopter; tap Phyrexian Tower and sacrifice Ornithopter for {B}{B}. Divest to discard Kinnan, Bonder Prodigy, then Claim to return Kinnan, Bonder Prodigy to the battlefield. Mox Amber and tap it for {U}{U}. Myr Scrapling, then Of One Mind to draw 2.
(Mox Amber (previous Mox is sacrificed due to the legend rule), tap for {U}{U}, Of One Mind)×3, ending up with 3 blue mana and 2 cards in hand. Scrap Trawler. Sacrifice Myr Scrapling targeting Scrap Trawler, triggering Scrap Trawler to get back a Mox Amber; cast and tap it for 2 mana. Chromatic Sphere and sacrifice it, getting 2 mana because of Kinnan, Bonder Prodigy, drawing a card, and getting back Mox Amber with Scrap Trawler, for another 2 mana. Cast and sacrifice another Chromatic Sphere, but this time hold on to the returned Mox Amber. (Now 4 mana – make some of it white – and 1 new card in hand.) Teshar, Ancestor's Apostle. Now loop: cast Mox Amber and return Myr Scrapling to the battlefield with Teshar, Ancestor's Apostle, tap Mox Amber for 2 mana, then sacrifice Myr Scrapling and get back Mox Amber with Scrap Trawler.
2 turns, 9 cards
Turn 1: Forest, Gilded Goose, Mox Amber.
Turn 2: Aether Hub. Tap the lands and spend the Energy for Kinnan, Bonder Prodigy. Tap Gilded Goose and sacrifice the Food it created for {R}{R} to cast First Day of Class. Tap Mox Amber for {G}{G} to cast Incubation Druid, which gets haste and a +1/+1 counter. Tap it for {U}{U}{U}{U}. Cycle Vizier of Tumbling Sands to untap Incubation Druid, with {U}{U} left, and tap it again for {W}{W}{W}{W}. Now cast Gauntlets of Light on Incubation Druid, and repeatedly untap and tap it to get infinite mana.
Also 2 turns, 9 cards
Turn 1: Forest, Gilded Goose, Mox Amber.
Turn 2: Island. Tap the lands for Kinnan, Bonder Prodigy. Tap Gilded Goose and sacrifice the Food it created for {R}{R}, and tap Mox Amber for {G}{G}. Spend {R}{G}{G} on Xorn and the remaining {R} on Strike It Rich, producing two Treasures. Sacrifice the Treasures for 4 red mana and spend 3 to cast Strike It Rich again by flashback for another two Treasures, and sacrifice those, for a total of 5 mana. Cast Goldspan Dragon and have it attack, producing two Treasures that can now make 3 mana each, for a total of 6 mana. Cast Conviction on Goldspan Dragon, producing another two Treasures, and return it to hand with its ability, and repeat for infinite mana.