In order for such a game actually to be played, both players would have to work at it, so I'm not so sure about the "aimlessly as possible" part. Neither the 50-move rule nor the 3-move draw force a draw. Both players could refuse to ask for the draw.

However, if you're really asking that the opportunity doesn't come up, then you've definitely got some limitations. The number 49 will likely be a factor of a significant portion of the desired bound. Now, the maximum number of pawn moves possible before a capture is necessary is equal to 32 (say, one side's pawns moving as slowly as possible: 4 positions before hitting the opponent's pawns times 8 pawns). We'll assume that both players can easily avoid the invocation of the 3-move rule up to this point. So now you're already up to 32 x 49 moves possible. You would then have to have 8 captures to move all of one side's pawns to the last rank for promotion, which would, incidentally, allow the other side's pawns to promote. However, you'd have to have at least one pawn of one side being captured. So you now have the full 6 ranks x 15 pawns = 90 pawn moves, along with 8 captures. You would then have to continue capturing pieces, since there would be no pawns left. There are 30 - 8 pieces to capture. So, in total, so far, you have (90+30)49 = 5880 moves. Let's assume that the last of the 30 pieces captured was a queen, to allow for the possibility of checkmate at all times so far. Now you're down to king versus king. Checkmate and stalemate are both impossible, so I think the rules of chess would say that a draw was forced on you. If not, you could, if you wish, go into the number of possible positions occupied by both kings, which is equal to

4 x 60 (white king occupies a corner)

+ 24 x 58 (white king occupies a non-corner edge square)

+ 36 x 55 (white king occupies an interior square)

= 240 + 1392 + 1980

= 3612 possible positions for the kings. Therefore, to the 6174 moves already mentioned, you could add another 3612 + 3612 + 1 = 7225 moves for a grand total of 13105 moves. But that last one is debatable.

Keep in mind that I am also assuming that, for instance, in this incredibly boring endgame with two kings, it is possible to get all the positions listed. It may not be possible because of the rules of the game. So I merely am attempting an upper bound on the number of moves.

These are my thoughts. They could be way off, but take them for what they're worth.