Go means ( define:go in google )
move from one place to another; travel.
"I really must go"
synonyms: leave, depart, take one's leave, take oneself off, go away, go off, withdraw, absent oneself, say one's goodbyes, quit, make an exit, exit;
May I get some water ? ( without going somewhere how will I get water. Water will not come to me )
So, May I go get some water ? suggests that although it is implicit in get ( to go ), adding another go to it ensures you are communicating ( very clearly ) that look i have to go ( take trouble to move ) to get you some water but i will go.
A very lazy view.