Honestly, I used to think a 3v3v3 map was possible too, but once you start delving into balancing everything, you understand the 1 huge conceptual flaw about it : backdoor. Basically, there is no incentive to fight.
Let's say there's team A, B and C. For this example, exceptionally, they are level 11 and everyone has the exact same money and experience as well as the same skill level. So A sees that B is trying to catch C off guard because C's ADC is farming a camp far from his team. B initiates on the ADC and kill it with 4 members. We now have the situation :
A : 5 Alive near B's base.
B : 1 Alive near B's base, 4 90% hp near C's base
C : 4 Alive near C's base.
So now, C clashed on B, trying to save their dude and raging about how he was so off position and that his mom sucked because he was a afro-canadian and blah blah blah, but the point is, C initiates on B.it's still a 4v4 with B missing some HP.
Now B has 2 options, try to peel (and possibly lose 1-2 champs) to defend from A. Or fight the fight and possibly win (since C is missing a key player, and the 5th B could join in), but leave A to backdoor their base endlessly.
In the same way, C has 2 options, actually chase B down, or wait for their 6th to come back up, depending on what A is doing (if A comes, C leaves. If A backdoors, C chases).
And finally, A has 2 options, Try to catch both teams and score a lot of kills, or try to backdoor B.
So let's see in what position they are :
C : neutral (fight an unsure fight or do nothing)
B : Lose-Lose (get backdoor'd or get killed)
A : Win-Win (get kills or backdoor)
So basically, B is punished for initiating, C hasn't really lost a lot although they got caught off position because they played badly and A got rewarded for not doing anything.
Even if all 3 don't ever hard engage onto someone, the one who will always come ahead is the one who didn't take part in that fight...
Another reason why 3 teams wouldn't work is because of how strong AoE becomes... Basically, Karthus is an instant ban, and so is Teemo, Kog'Maw, Lux, Garen, Wukong...
And last but not least, the reason why it's a non-functional concept is because of the multiple 2v1 situations... basically, KSers like Katarina, Darius, Twitch, Nidalee, etc. becomes must pick, because you don't want the 3rd team to get the kill. And that can be extended to the victory scenario... Basically, if you win by killing a nexus, then Amumu becomes a must pick... stand by nexus, wait till it gets low, Flash + Ult and last-hit while they can't AA... If it is a last-man standing, then you are even more penalized by fighting (because the 3rd party will roughly have the game in its hands.
If there ever is a map such as a SR-like 5v5v5, I think the winning team is the one with the hardest late game, whatever their early game is like... They can't die early or the team that killed them will die... So basically, I run Amumu, Malphite, Vayne, Kogmaw and Karma
Other fundamental flaws from 3-sided fights includes : Mid lane, all minions will always be stuck in the middle until a team dies. Basically, 1 team's minions can never outpush 2 teams'
If you REALLY wanna create such a map, you would need to have something like a central buff that gives invulnerability to your base for like 5-8 minutes with a respawn timer of ~75% that duration. But then again, it simply favor AoE/ks further because there will be 15 persons fighting over 1 buff. So the next option is smaller buffs that grant it to the teams, with buffs stackings (something like 2m45 invulnerability with a 5m respawn time. but then you have a team that can simply snowball as hard as he wants, with no way to denies it.
Basically, to jot it down into simple statements, for a 3-sided battle,
1) You need an incentive to attacking.
2) Because of #1, you need a reliable way to defend yourself.
3) You need a critical way to counter #2, so that by outplaying your opponent, they can actually throw and you can create a comeback.
4) You need a way to prevent/promote ganging up, depending on the situation. (but in most cases, it'll be prevent).