Mirage Mirror
Oracle Text
: Mirage Mirror becomes a copy of target artifact, creature, enchantment, or land until end of turn.
Card Rulings
7/14/2017 Once Mirage Mirror’s ability resolves, it no longer has that ability.
7/14/2017 Mirage Mirror copies the printed values of the target permanent, plus any copy effects that have been applied to it. It won’t copy counters on that permanent or effects that have changed its power, toughness, types, color, or so on. Notably, it won’t copy effects that made the target permanent become a creature.
7/14/2017 If Mirage Mirror copies a permanent that’s copying something else, it will become whatever the target is copying.
7/14/2017 If you activate Mirage Mirror’s ability multiple times in a turn in response to itself, then each time one of those abilities resolves, it will overwrite whatever Mirage Mirror is copying. Mirage Mirror will wind up as a copy of the permanent targeted by the last ability to resolve. When the turn ends, all instances of the ability will wear off at the same time.
7/14/2017 If an effect begins to apply to Mirage Mirror before it becomes a copy of another permanent, that effect will continue to apply. For example, if Mirage Mirror is activated twice in response to itself targeting first Rampaging Hippo then Frilled Sandwalla, the ability it has while it’s a copy of Frilled Sandwalla can be activated and its effect will continue to apply while Mirage Mirror is a copy of Rampaging Hippo.
7/14/2017 If Mirage Mirror becomes a creature the same turn it enters the battlefield, you can’t attack with it or use any of its abilities (if it gains any) unless it has haste.
7/14/2017 If Mirage Mirror becomes a copy of a legendary permanent you control, you’ll put one of them into your graveyard.
7/14/2017 If Mirage Mirror becomes a copy of an Aura, it’s put into its owner’s graveyard unless it’s somehow attached to an appropriate object or player already. If it becomes a copy of an Equipment and is attached to a creature, it’ll become unattached when it becomes a non-Equipment artifact again.
You can attack with it the next turn Mirage Mirror has entered the Battlefield.
Although it would no longer be called Mirage Mirror so I don't know if each trigger would resolve or just the first. Of course I can't imagine too many scenarios in which having it change what creature it is multiple times in a row would be helpful.
So you can activate the ability multiple times on the stack and then as each one resolves you can then do other things before the next change happens. You are right, the uses are limited but they do exist .. something like tap it as one creature and then have a Shadowmoor card with an untap cost untap it.
Doubling Seasons your turn creature to block their turn.
701.37a To exert a permanent, you choose to have it not untap during your next untap step.
Because it says "permanent" in the rule, my guess is exerting after copying a creature with exert will stop it from untapping.