Double battles (and their similar brethren of Multi Battles/Tag Battles) for the win! Starting with Pokemon Sapphire Version may have influenced me a bit, but they are so fun! Lots of strategy that gain viability in the Double Battle Mode that are practically useless in the Single Battle Mode. I also like doubling up the offensive on a single opposing Pokemon, as well as "area of effect" attacks like Earthquake that affect the whole field. I also relish that moment when the match-up becomes "2 Pokemon vs. 1 Pokemon", and the "flow/continuity" in battle that occurs when a foe Pokemon faints by your first Pokemon's attack and has to be replaced by another Pokemon, only for the newly-switched-in foe Pokemon to become helplessly susceptible and fall prey to your second Pokemon's attack in the same turn! (well, at least going by Generation 3 mechanics..) Things like Tailwind and Skill Swap are much more useful and appreciated during Double Battles, too..
Triple Battles and Rotation Battles seem a bit overwhelming despite my love of Double Battles; I'm surprised we even made it that far (not to mention "1 vs. 5" Horde Encounters)... Inverse Battles seem like a nice variation, as do Sky Battles (although the list of eligible Pokemon still needs revamping).
As for future battle variations, all I can think of is "Underwater Battles", which Pokemon OmegaRuby/AlphaSapphire should have had.. I hope that all future Battle Towers (and the Battle Dome, Battle Factory, and Battle Palace, if they return) incorporate the Battle Modes of Single Battle, Double Battle, Multi Battle, Triple Battle, and Rotation Battle, with options being available so that you can change the battle flavor (aside from "Standard" Battle, Sky Battle and Inverse Battle could be selectable 'modifiers'), so that things like "Inverse Sky Multi Battle" or "Sky Rotation Battle" are possible.
Pokemon Emerald did a good job of integrating Double Battles and a Multi Battle; hopefully, future Pokemon games can integrate more Double Battles, Triple Battles, and Rotation Battles (along with Inverse Battles and Sky Battles) into the main adventure.
I also liked the Hoenn Battle Arena's "Set-KO Tourney" event, although I can see how that battle variation could be limiting for certain Pokemon.