So is it possible to copy eternal dominion on the stack, to create multiple instances of it on the upkeep? If I copy it a lot of times as it is cast, will each of those copies copy themselves again on the upkeep?
When you cast Eternal Dominion you can use cards like Twincast or the ability of Echo Mage to copy it. If you copy at this point each copy will "go off" on your upkeep. If you use the mage to copy a recurrance of the epic it won't repeat because the copy made by epic doesn't have Epic.
A spell with epic keeps you from casting spells for the rest of the game only when that spell resolves, not when it's cast (C.R. 702.50a, 608.2c). Like almost any other spell, a spell with epic can be "responded to" by Twincast or other spells (but see this thread) (C.R. 117.7). Like any other spell, a spell with epic has no effect unless and until it resolves (C.R. 609.1, 608.2c). See also this thread.
Note that copying a spell copies all its normal abilities, including epic, unless the effect states otherwise (e.g., the epic keyword's own effect) (C.R. 706.2, 702.50a, 207.1).
EDIT (Apr. 20, 2022): Edited, including because some rules were renumbered in the meantime.
So is it possible to copy eternal dominion on the stack, to create multiple instances of it on the upkeep? If I copy it a lot of times as it is cast, will each of those copies copy themselves again on the upkeep?
Yes, and yes.
Each copy you create via Twincast and friends will re-copy itself each upkeep until the end of the game, thanks to their EPIC ability.
So, one Eternal Dominion plus three Twincast gets you four Eternal Dominion each upkeep, for the rest of the game.
Card tags fixed. -Madmage
Note that copying a spell copies all its normal abilities, including epic, unless the effect states otherwise (e.g., the epic keyword's own effect) (C.R. 706.2, 702.50a, 207.1).
EDIT (Apr. 20, 2022): Edited, including because some rules were renumbered in the meantime.
Each copy you create via Twincast and friends will re-copy itself each upkeep until the end of the game, thanks to their EPIC ability.
So, one Eternal Dominion plus three Twincast gets you four Eternal Dominion each upkeep, for the rest of the game.
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules