Declare Balan as an attacker.
Holding priority, activate Balan by paying {1}{W}, attaching Assimilation Aegis to it.
Assimilation Aegis' second ability triggers, causing Balan to become a copy of Medomai the Ageless.
Deal combat damage with Balan.
Balan triggers, causing you to take an extra turn after this one.
Move to your next turn.
Activate Assimilation Aegis' equip ability by paying {2}, attaching it to any other creature you control.
Assimilation Aegis' second ability triggers, causing the creature to become a copy of Medomai the Ageless.
Repeat.

Infinite turns

Medomai the Ageless

Assimilation Aegis

Balan, Wandering Knight

Declare Balan as an attacker.
Holding priority, activate Balan by paying {1}{W}, attaching Dinosaur Headdress to it and causing Balan to become a copy of Medomai the Ageless.
Deal combat damage using Balan.
Balan triggers, causing you to take an extra turn after this one.
Move to your postcombat main phase.
Activate Dinosaur Headdress' equip ability by paying {2}, attaching it to any other creature you control.
Repeat each turn.

Combos with "Balan, Wandering Knight"