What do you call a verb phrase with no subject, like a description of purpose: "to exact revenge" or ability: "juggle cats while tap dancing"?
If you're talking about these phrases as part of a larger structure, and you're using a theory that has "little v", you can call these (big) VPs:
If you're talking about these phrases on their own, though, they can be complete clauses (TPs or IPs or what have you), just without pronounced subject. Infinitive clauses like "to exact revenge", for example, are generally said to have an invisible subject called "PRO":
We know it's there, even if it's not pronounced, because it can e.g. bind reflexives ("[the goal is] to see myself"). Likewise in imperative clauses ("look at yourself"). So you could call these "clauses with PRO subject", or "clauses with unpronounced subject", but I'm not sure if that's a category that's generally used.
(These differ, of course, from the famous "English sentences without overt grammatical subject", like "bless you" or "damn you". These seem to actually lack a subject entirely, rather than just having an invisible one like PRO.)