When Kitchen Finks dies, its persist ability triggers, returning it from your graveyard to the battlefield with a -1/-1 counter and a vigilance counter on it.
When Kitchen Finks enters the battlefield, it triggers, causing you to gain 2 life.
Activate Tayam by paying ({3} magic symbol) , removing a -1/-1 counter from Devoted Druid, and both counters from Kitchen Finks.
Holding priority, repeat steps 3-5 one additional time.
Activate Phyrexian Altar by sacrificing Kitchen Finks, adding one mana of any color.
Resolve Tayam's ability from step 5, causing you to mill three cards and returning Kitchen Finks from your graveyard to the battlefield.
Repeat.
Results
Infinite colored mana.
Infinite death triggers.
Infinite ETB.
Infinite lifegain.
Infinite lifegain triggers.
Infinite LTB.
Infinite sacrifice triggers.
Infinite self-mill.
Put all permanents with mana value three or less from your graveyard and library onto the battlefield.