The highest I've gotten so far is in the 16000's. But, I do seem to be getting better. @girlgonemild mentioned (in person) she keeps all the high numbers to one side and that helped me get the 1024 tile.
An acquaintance of mine is an AI professor at MIT and wrote up a custom script:
So, it's cheating, but it's a pretty impressive form of cheating. Apparently it gets crazy hard to get each additionally higher number: "Once you have made 8192, it's essentially just as hard as that to make another 4096, and then just as hard to make 2048."
So making 4096 is much much more than twice as hard as making 2048. Nice job, @girlgonemild.
I found that it helps if you move in a snake like pattern. I keep all of my largest tiles on the bottom with the biggest number on the right. The next row will have the biggest numbers on the left, allowing me to constantly chain the smaller numbers into the bigger ones. I think this will be the last one I post. There just wasn't much room to move around once you get the 8192 tile.