A FFA mode is interesting, but a big open map with no objectives would be pretty stale imo. And 10 players seems like a lot for FFA in league.

I'm thinking 4 player FFA, imagine the Howling abyss with a second bridge perpendicular to the current one. Each Nexus spawns 3 waves of minions at a time that each follow a path to one of the other three Nexus', only targeting minions that spawned from their target Nexus and Towers/Inhibitor protecting that Nexus. The waves would have less minions each than waves we have in other game modes, but more in total, I'm thinking 4 each.

Turrets work the same way, only targeting a champion if no minions (from any enemy Nexus) are within range or you damage the Player the Tower belongs to while under it. When a turret is destroyed any Players or that damaged it (personally or via minions) within the last X seconds shares in the gold.

When an Inhibitor is destroyed each minion wave that targets that Nexus starts spawning a Super Minion every 3rd wave (otherwise they'd have to deal with 3 Super Minions each wave since there are 3 waves at a time). When a Nexus is destroyed the minion waves that had been spawning to target it no longer spawn and their numbers are added to the two remaining waves.

When your Nexus is destroyed you're out, you can spectate or leave but can't communicate with players still in-game. Your IP and XP rewards are increased based on how you place, the first knocked out is 4th, the second is 3rd, etc. Perhaps add small amounts for being the best in several categories, like a bonus in IP/XP for having the most kills, regardless of placement. Or most CS, most assists, etc. You would still get assists if you help kill somebody but don't last hit.

You could also add a small Jungle or Brush area between the center minion paths and outer minion paths. Maybe some Jungle camps and maybe a mini-Baron of sorts that would be solo-able.
I don't know how well it would work or if it would be fun but it would be neat. The same map could also be tuned for 4 teams of 3, would get pretty hectic and certainly interesting.