Yes. In fact, it's not only possible but very common.
Most people who complete all the levels in the game have not bought anything at all.
Alphabetty will take some skill on your part, to spot those words and tactically create them so you can make the most of the boosters and bonuses the long words will deliver.
Playing each level will not only teach you the skills you need, but will train you to get better and better at the puzzle aspect, so working your way forward should arm you with the tips and tricks you need.