I’ve seen in a text しています and 行っていて. I figured the first one was from the する verb and the second one was from 行く. I looked both verbs in a dictionary app and could only find that both’s first parts are from the positive informal -te form, but couldn’t figure out where います and いて came from. Are these other verbs? Or something else that adds something to the verbs?
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https://www.tofugu.com/japanese-grammar/verb-continuous-form-teiru/
行っていて means “is going and (something)”
Both of the endings are conjugations of いる. It can be conjugated just like a “regular” verb.
A fun application of this is with the negative form of verbs. You could say 行く to say you’re going, or you could say 行かなくない to say you’re not not going (double negatives are legal in Japanese). But in reality you wouldn’t hear the latter very often.