My lawn is slowly being overrun by thyme. My neighbors yard is already about 70% thyme and that is where it has been slowly creeping over from since we landscaped in 97.
People who are lawn snobs are appalled when I tell them this but I hope it overruns my whole yard. He hardly ever cuts his lower yard, far from the house as he likes the purple flowers but even then it never grows more than a few inches. If you keep it cut short, it just thickens into a soft, gentle carpet that's easy on the feet and smells great when you walk on it. Even then, it hardly ever needs mowing. Just when the flowers start to form.
If I had money, I'd reseed with nothing but thyme.