The throw-aways are the best way to go. You can add money when necessary, do so anonymously, and they're cheap. Just make sure to register it using an anonymous email address, such as Yahoo, and pay for the phone and additional time in CASH. I go so far as to use a second-hand laptop I purchased in cash at a yard sale on public WiFi (restaurants, hotels, etc.) to add time to the phone and to register it. Overkill? Maybe, but why take chances...

they asked for
an address: I improvised
a company: I always wanted to work at Boeing as a rocket scientist

no ID

I have about 1000 minutes...  I renew every 90 days without letting it expire so the minutes roll over.   I buy my airtime anywhere for cash.