Turn 1; 7 cards (plus 12 additional cards are required to be in library/sideboard for this to be deterministic); with a constraint on deckbuilding; Deterministic and produces any amount of all colors of mana!
Starting hand: Any leyline, any white-producing untapped land, Fragment Reality, Mox Amber, Spellchain Scatter, Faithless Looting, Underworld Breach.
Play pattern: Start with Leyline on the battlefield. Play a land. Cast Fragment targeting Leyline. Only creature in Library this can hit is Birgi, God of Storytelling. Cast Mox Amber. Cast Spellchain Scatter. Cast Underworld Breach. Cast Faithless Looting, discarding both cards drawn off of it. [Spellchain's effect conjures a second copy of Faithless Looting into hand.] Cast conjured copy of Faithless Looting.
Hand is now empty.
Graveyard is now: Fragment Reality, Spellchain Scatter, two copies of Faithless Looting, and four other arbitrary cards from our library.
Escape Spellchain Scatter by exiling Fragment Reality and two other cards.
Escape Spellchain Scatter again by exiling one of the Lootings and the two other arbitary cards, leaving the second Looting in the graveyard.
This conjures a copy of Spellchain Scatter into our hand, which begins the infinite loop: every copy cast generates a mana off Birgi and conjures an additional copy of Spellchain Scatter into our hand.
We can now generate an arbitrarily high storm count and an arbitrarily large number of cards in our graveyard.
Once we reach our desired arbitrary number, we can escape the other Faithless Looting from the graveyard over and over again until we discard into our graveyard a second Mox (infinite Red mana off Birgi triggers and legend rule). To keep this deterministic and prevent the risk of exiling all 3 remaining moxen before starting the loop, we also run 2 copies of Tormod's Crypt (also generates infinite red mana).
Now that we have infinite red mana, we cast Faithless Looting over and over until we discard a mana filter (examples: Energy Refractor, Crossroads Candleguide, Mimic, Signpost Scarecrow, Urn of Godfire). To prevent the possibility of exiling all these, we run one copy plus four copies of Wish with a second filter in the sideboard.)
If a wincon is also desired, 1 Grapeshot (plus one more in the sideboard) will do the trick.
(old answers archived below)
Turn 1; 7 cards (+11 more from library); with a constraint on deckbuilding, no additional luck needed
Start with Leyline of Sanctity (or any other Leyline) on the battlefield. Play Plains. Cast Fragment Reality targeting the Leyline. The only creature in the library with mana value three or less is Birgi, God of Storytelling (comes into play tapped).
Cast Mox Amber; get {R} from Birgi's trigger, tap it for {R} (total in pool: {R}{R}).
Cast a second Mox Amber; get {R} from Birgi's trigger, keep the new one when legend rule applies, tap it for {R} (total in pool: {R}{R}{R}{R}).
Cast (a one-of in the deck) Tormod's Crypt; get {R} from Birgi's trigger (total in pool: {R}{R}{R}{R}{R}).
Cast Indomitable Creativity with x=2 targeting Crypt and Mox. Get {R} from Birgi's trigger. The only two creatures in the library are Razaketh the Foulblood and Hornet Queen. Using Razaketh's ability, sacrifice Hornet Queen and its four tokens to search your library for: Strike It Ritch, a second Strike It Ritch, Pact of Negation, An Offer You Can't Refuse, and Gleeful Demolition.
Cast Strike It Ritch; get {R} from Birgi's trigger, (total mana available: {R} in pool and one treasure token).
Cast Strike It Ritch, get {R} from Birgi's trigger, while it's on the stack, cast Pact of Negation, get {R} from Birgi's trigger, while it's on the stack, sacrifice a treasure to cast An Offer You Can't Refuse targeting Pact, get {R} from Birgi's trigger (total mana available: {R}{R}{R} in pool and three treasure tokens).
Cast Gleeful Demolition, targeting a treasure token (total mana available: {R}{R}{R} in pool and two treasure tokens).
Using Razaketh's ability, sacrifice the three Gleeful tokens to search up from your library: Strike It Rich, Wish, and Reckless Charge.
Cast Strike It Rich, get {R} from Birgi's trigger (total mana available: {R}{R}{R} in pool and three treasure tokens).
Cast Wish, get {R} from Birgi's trigger (total mana available: {R} in pool and three treasure tokens).
Using Wish's effect, play from sideboard Selvala, Heart of the Wilds, get {R} from Birgi's trigger (total mana available: {R} in pool and one treasure token).
Cast Reckless Charge targeting Selvala, get {R} from Birgi's trigger (total mana available: {R} in pool and one treasure token).
Tap Selvala, paying {G} from the final treasure token to get 8 mana in any combination of colors off Razaketh's power.
Using Razaketh's ability, sacrifice Birgi to find from your library and cast High Alert. Repeatedly untap and tap Selvala to make infinite mana.
Once you have whatever amount of mana you fancy, use Razaketh's ability to sacrifice Selvala to retrieve another copy of Wish, and get any win condition you please from your sideboard.
(If you're not concerned with generating infinite mana, rewind to the casting of Indomitable Creativity and find Griselbrand and Queza, Augur of Agonies for lethal damage.)

Turn 1; 8 cards; with a constraint on deckbuilding, no additional luck needed, infinite red mana
Start with Leyline of Sanctity (or any other Leyline) on the battlefield. Play Plains. Cast Fragment Reality targeting the Leyline. The only creature in the library with mana value three or less is Birgi, God of Storytelling (comes into play tapped).
Cast Mox Amber; get {R} from Birgi's trigger, tap it for {R} (total in pool: {R}{R}).
Cast Underworld Breach; get {R} from Birgi's trigger (total in pool: {R}).
Cast Spellchain Scatter; get {R} from Birgi's trigger (total in pool: {R}).
Cast Spellchain Scatter; get {R} from Birgi's trigger (total in pool: {R}); conjures a new copy of Spellchain Scatter into hand; repeat this last step an abitrary number of times, filling your graveyard with an arbitrary number of cards (and generating an arbitrary storm count).
For infinite mana, cast either a 2nd Mox Amber (one goes to the graveyard via legend rule) or a Tormod's Crypt (or other 0-mana permanent that sacrifices itself), generating {R} off Birgi, repeat an arbitrary number of times by casting the card from your graveyard by paying its escape cost of three cards—remembering that we have an arbitrary number of conjured cards in our graveyard.