Fishdom is absolutely free to play for everyone. All levels are beatable without spending real money. You can get some in-game items using in-app purchases to make things easier if you want, but you certainly don't have to.

However, if you're worried about in-app purchases, you can adjust your device's settings (you can block purchases or password-protect them). That is especially useful if you have children who play on your device. You can find more information about restrictions on in-app purchases in the relevant article.