Cast Dramatic Reversal by paying {1}{U}.
Holding priority, activate Izzet Guildmage's first ability by paying {2}{U}, copying Dramatic Reversal.
Resolve the Dramatic Reversal copy, untapping all nonland permanents you control.
Activate all nonland permanents by tapping those, adding at least {3}{U}.
Repeat from step 2.

Infinite colored mana

Izzet Guildmage

Dramatic Reversal

Cast Lava Spike by paying {R} and choosing to pay {1}{R} and reveal Desperate Ritual from your hand, adding Desperate Ritual's effects onto Lava Spike, with Lava Spike targeting an opponent.
Cast Desperate Ritual by paying {1}{R}, adding {R}{R}{R}.
Activate Izzet Guildmage's second ability by paying {2}{R}, copying Lava Spike, targeting an opponent.
Resolve the copy of Lava Spike, adding {R}{R}{R} and dealing three damage to an opponent.
Repeat from step 4.

Infinite damage

Infinite magecraft triggers

Lava Spike

Desperate Ritual

During an opponent's turn, cast Reset by paying {U}{U}.
In response, activate Izzet Guildmage's first ability by paying {2}{U}, copying Reset.
Allow the Reset copy to resolve, untapping your lands.
Activate those lands by tapping them, adding at least {2}{U}.
Repeat from step 2.
If your lands can produce at least one additional mana, this combo produces infinite mana.
Once you have infinite mana, you may activate Izzet Guildmage's abilities any number of times.

Infinite copies of instant and sorcery spells on the stack with mana value 2 or less that you control

Infinite mana lands you control can produce

Infinite untap of lands you control

Reset

Cast Turnabout for {2}{U}{U}.
In response, Cast Twincast targeting Turnabout for {U}{U}.
In response, Activate Izzet Guildmage's ability for {2}{U} targeting Twincast.
Resolve the Izzet Guildmage's activated ability to copy Twincast targeting Turnabout with the copy of Twincast.
Resolve the copy of Turnabout, created by the copy of Twincast to untap your lands that can produce at least {3}{U}.
Repeat.

Infinite colorless mana

Turnabout

Twincast

Cast Manamorphose by paying {1}{R/G}.
Holding priority, activate Izzet Guildmage by paying {U}, copying Manamorphose.
Resolve the Manamorphose copy, adding two mana of any color and drawing a card.
Repeat until you have one card remaining in your library.
Resolve the original Manamorphose, adding {U}{U} and drawing a card.
Cast Thassa's Oracle by paying {U}{U}.
Thassa's Oracle enters the battlefield, causing you to win the game.

Infinite card draw

Near-infinite colored mana

Near-infinite magecraft triggers

Win the game

Zirda, the Dawnwaker

Manamorphose

Thassa's Oracle

Cast Manamorphose by paying {1}{R/G}.
Holding priority, activate Izzet Guildmage by paying {U}, copying Manamorphose.
Resolve the Manamorphose copy, adding {1}{U} and drawing a card.
Repeat from step 2 until you draw from an empty library, winning the game.

Laboratory Maniac

Cast Manamorphose by paying {1}{R/G}.
Holding priority, activate Izzet Guildmage by paying {U}, copying Manamorphose.
Resolve the Manamorphose copy, adding {1}{U} and drawing a card.
If Jace is on the battlefield, repeat from step 2 until you draw from an empty library, winning the game.
Otherwise, repeat from step 2 until you have one card left in your library.
Resolve the original Manamorphose, adding {U}{U} and drawing a card.
Cast Jace from your hand by paying {1}{U}{U}{U}.
Activate Jace's first loyalty ability, causing you to draw from an empty library and win the game.

Combos with "Izzet Guildmage"