Yes, you will hit a level cap of 13 (or 18 when you verify your account) if you don't open your wallet, and be out-geared. It costs only $9.99 to purchase every campaign at the moment, excluding PvP maps.
You can also receive platinum for free by completing tapping the GET FREE PLATINUM link in your character selection screen (near the top). If you're on an iOS device it's entirely possible to finish enough offers to get every single campaign without spending any cash whatsoever.
Simply download the free applications on the Tapjoy offer list, open and run each application, and check to see if your Platinum was received (you have to log off and log back in to see the received Platinum). Delete the application after you received the Platinum and keep doing offers! Spending roughly 10 minutes of your time completing these free offers will grant you enough platinum to buy one new campaign map, which is certainly well worth it considering a campaign map delivers hours and hours of gameplay.
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