I’m not sure if “aggr-” is a prefix but I can see some words starting with it. like:
I’m here to ask if it has some meanings or they are all started with “aggre-” by chance.
No, aggr- is not a prefix. Latin had a prepositional prefix ad- that turned into ag- before g-. In the course of French’s development from Latin, the preposition ad and the prefix ad- lost their final consonant sound. But French words formed with that prefix were sometimes spelled with doubled consonants in imitation of Latin practice (but not always: see fdb’s answer to Why does “agree” have only one “g”?).
The aggr- sequence in all of the words that you list was originally formed by adding the Latin prefix ad- (ag-) or the French prefix a- (ag-) to bases starting with gr-. Aggrieve has the same etymological base as aggravate (the ie in aggrieve is from a French sound change).
The bases of the other words (grand-, greg-, gress-) go back to separate Latin roots. Although their pre-Latin etymologies seem a bit uncertain, I don’t know of any proposals for etymologies that connect any of these roots.