Looks like Paulownia tomentosa, an invasive tree from elsewhere. Next time you cut it down, brush a little Brush Killer on it (Brush B Gon).

Here is a link that might be useful: Paulownia tomentosa

This is paulownia tomentosa to compare your leaves against.

As you can see the pop up in places they weren't intended to be as they are very invasive and they roots can also spread unbelievable distances.

I had one in my back yard when we lived in zone 8. The thing grew extremely rapidly and was a pain to kill. The advice that esh ga gave you is really good advice. We cut ours down a couple of times, then finally the last time we cut it down, we used the brush killer. It finally gave up and never came back.

Beverly

Some of my fondest childhood memories include throwing the seed pods of this tree at friends in fall when they are so sticky they cling to clothing. While it is quite invasive, it does have beautiful blooms, then golden seed pods that eventually turn to brown, sticky pods full of flufy seeds that can germinate in a crack in concrete if needed. It's kind of like a Kudzu Tree! If left alone it will shade out all other plants in the area. The wood is prized by some for it's ease in carving, by the way.