That's a lot of limitations. I take my dog every time I leave the house! Maybe cheat and also take my passport :p
That said, probably somewhere like Aus or NZ...
It'd have to be extremely public, preferably during a live broadcast sporting event or something. To generate buzz and news. Hoping that's enough for them to let me stay and make some money :p
Or challenge someone to a race. Or an escape challenge make some money. The problem here I is getting it set up in time.
You can take your passport, but it's going to interesting when you try to leave the county with no record of entering. Getting back into your home county will be fun too.