Yeah, I once had a stray show up in my house. On the kitchen counter, to be exact. We had cat doors for the two existing cats in the house, and Teddy just showed up one day. I spent the next month chasing him out of the house every time I found him inside, on the porch, or on the deck. Then I gave up.
He stayed for about two years, through his kidney failure, $2K in vet bills, and twice daily medication. He finally succumbed a few days after Christmas 2001.
I miss him.
Keep the stray.