>How much can you play?
As much as you like. First playthrough is about getting to Glalie before Team Aqua and Magma. You get a "mission" with every playthrough after that, all of them are pretty simple. Beat three weak trainers, one strong trainer, find a Skitty/Poochyena, talk to a kid who's usually behind a trainer, etc.
>Is there a play count until it expires?
I'm not entirely sure. It kept track until the tenth time, now it just says "lost track". It may reset tomorrow.
>Is the game limited in anyway?
You are in Mossdeep City, all the buildings are locked, and the only way you can travel is by doing that playthrough's "mission" with Steven. You have one fully evolved starter at first, any playthroughs after that you have all three and Glalie, all of which have their Mega Stones.
It's really a teaser more than anything. NPCs appear at random with each playthrough, talking about fossils, contests, Sea Mauville, etc.
On a related note, anyone figured out what's in front of Steven's house? It's an invisible thing blocking the door.
