This process should keep going, to keep you growing.
Is the use of "keep you growing" grammatical?
The construction is perfectly fine. to VERB clauses may be used this way to indicate purpose or goal.
You may want to rethink your main clause, however. I suspect that the ‘process’ you are talking about is something which is done by the person you are talking to. If that is the case, you should make that person the subject of the sentence, so that you are urging him to “keep up the good work”: