You’re not playing them, you’re just attacking with them.
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Oh my god you’re telling me the Hyperion infinite is only possible IF YOU’RE NOT THE ONE PLAYING THE HYPERIONS?! THAT. IS. AMAZING. 
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Your are playing some arrangement of past/truth/law, and your opponent is playing Present/X/X
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Or just Sanatorium into two Warp Gate Disciples for two real guys for 3 gold. It would potentially be a decrease in number of swings but in would be an increase in number of real bodies. Either way, it sounds like a fun thing to try.
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Okay BONUS ROUND!
What is the maximum amount of damage you can deal to an opponent’s base if you attack only with Hyperions? What’s the solution assuming “before your opponent takes a turn” to allow for Double Time shenanigans? What’s the solution for “in a single turn?”
Again, following all Codex rules, but assuming the most ideal legal set-up you can imagine. Also for this puzzle, assume that your opponent does not die when their base hits zero (i.e. you can overkill).
Winner Winner Chicken Dinner: EricF provides a modified version of his earlier solution!
Oh, I was about to calculate how much you could heal their base to be able to attack more than 20 times.
The base solution is “infinite” by moving unlimited +1/+1 runes from a Drill Sergeant (accumulated over however long is needed) onto a Hyperion and attacking once.
Also, my previous solution still works
Summary
: just kill off your own mirror Hyperion with an Archon rune each time it attacks.
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Are we in a 1-v-1 or can the solution be based on a 5 player free-for-all?
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If only out of pure curiosity please tell me how the 5 player free-for-all solution differs! 
Well, if you ban anything that goes “infinite” then the solution differs (because you can Assimilate a different number of cards from other players).
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SUPER SECRET DOUBLE BONUS ROUND: Same question as last time (i.e. most amount of damage using only Hyperions to attack) but your opponent is a CUNNING FOE who refuses to play Hyperions against you! In other words, what’s the best you can do when you are hampered by having to PROVIDE YOUR OWN HYPERIONS?
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I would never ban anything that goes infinite. 
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I might be misunderstanding, but are the mirrors attacking? If so, how? They don’t have haste, correct?
They’re copying Hyperions, which do have Haste.
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OMG, how foolish of me to overlook that! Thanks for the clarification!
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How can there be two Archons in play when both Tech II and Tech lab are locked into other specs? Is there some combination of spells/abilities I missed? Also, why is Macciatus necessary?
On a previous turn the Tech Lab for Past was destroyed and replaced with a new tech lab.
Macciatus is needed because otherwise the Insurance Agent would kill the mirror copy when he targeted it.
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I think I can still do infinite using my (kinda) own Hyperions:
I’m playing Law/Present/Past. My opponent is ?/?/Future. They have an Omegacron in their discard.
I have Tech 2&3 Past, and a Law Tech Lab
My hand contains Community Service, Seer, 2 Insurance Agents
I have a Remember (3 runes) and a Hyperion in play at the start of turn (I previously had a Present Tech Lab which I built over last turn)
I have 13 gold. (You only need 9, but 13 makes the loop easier to write)
Tech Ebbflow Archon
Remember fades (2 runes), bringing Ebbflow Archon into play
Summon Bigby [11g]
Play Community Service on Omegacron (6 runes) [6g]
Insurance Agent, insuring Archon [5g]
Insurance Agent, insuring Archon [4g]
– Start of Loop
Sacrifice Ebbflow Archon to Omegacron (5 runes) [20g]
Seer, removing a rune from Rememberer (1 runes), putting Ebbflow Archon into play from discard (8 runes) [19g]
Archon bounces both Agents and Seer (5 runes)
Insurance Agent, insuring Archon [18g]
Insurance Agent, insuring Archon [17g]
Seer, adding rune to Omegacron (6 runes) [16g]
Archon bounces Seer (4 runes)
Seer, adding rune to Rememberer (2 runes) [15g]
Archon bounces Seer (3 runes)
Attack with Hyperion
Summon Geiger, and max him, flickering Hyperion [9g]
Archon bounces Geiger (2 runes)
– Go to start of loop
ETA: I just realised after posting this that it won’t work like this infintely, because the Insurance agents make you draw cards also, so eventually you’ll need to shuffle, and you’ll redraw the Archon you just sacced. In this case, you play it from hand instead of using Rememberer. It will only happen once, as you’re only allowed to reshuffle once each main phase. You still get infinite Hyperions though 
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Despite the errata, pretty sure you still can. However it still requires an infinite gold loop and there are extra steps added to the setup to get around the errata.
[details=Summary]You are are Law / Truth / Other. Opposing specs include Truth / Future / Blood. (Note: particular objectives – such as the Hyperion challenge require more than one opponent)
Over an arbitrarily large number of turns where both players always have at least one gold: You use Community Service to gain control of an opposing Bloodrage Ogre. Opposing Quince makes a Mirror Glaxx with Quince’s maxband ability. You attack with Bloodrage Ogre ( Ideally there were prior interactions between armor and +1/+1 runes to make both the Ogre and the thing it attacks both have 0 attack, but if not you just assume an arbitrary previous number of castings of Community Service ) and gain control of the opposing Mirror Glaxx, thereby taking opponent (at least one) below their mirror illusion token limit. Opponent makes another Mirror Glaxx , you attack again with Bloodrage Ogre…etc…and over enough turns you can have as many mirror Glaxxes as you want in play, since you never used your own Quince’s ability to make any of them, his “limit” doesn’t apply.
Then you need an infinite gold loop in order to use your own Quince’s midband an arbitrarily large number of times to get enough gold to use your own Quince’s midband an arbitrarily large number of times to make all the mirror Glaxxes into mirror <>
[/details]
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That doesn’t work because the limit: 2 counts mirrors that you created, which have been stolen from you.
That’s not the answer I got earlier today in the rules thread.
And I’m pretty sure that if “limit: #” tracks prior creation regardless of control then Bird’s Nest and Harmony get all sorts of wacky in spec-mirror matches also involving any gain control effects.