But you can't. Not really. I guess it's location-dependant but if you live anywhere near a student campus or town there are literally tens of pokestops in a short 10min walk. My SO has got from level 1-11 without spending a penny, just by incorporating more pokestops into her little 'route' she has going. Added to that, the cooldown on a pokestop is 5 mins iirc, so you can pretty much milk them infinitely. If there's a stop in a pub for example, you'd be able to get 12 spins in the hour it takes you to catch up with your friends. Each spin tends to yield between 1-5 poke/greatballs.
Not sure what else is p2w, I was in the same boat as you until I saw how much progress my SO made without touching her purse.
I guess arguably the incense items in the cash shop are 'p2w', but you do get these through levelling up. Haven't seen any appear at pokestops yet. I've used 5 myself and all I've obtained through it are more Weedles and Pidgeys.