What is the difference between totally agree and completely agree? In other words, what is the difference in meaning between totally and completely in such combinations in conversations?
They mean the same thing, along with "fully agree" or "entirely agree". Either means agreement to the maximum possible and without any element of disagreement. "Totally" has picked up a connotation of sounding a bit ditzy but that seems to be on the decline. Otherwise, use whichever sounds better in context.