Alright, I didn't reroll 1s. So let's try this again:
http://invisiblecastle.com/find.py?id=973650
4d6 -> [6,6,6,5] = (23)
4d6 -> [1,3,2,4] = (10) -- reroll 1
4d6 -> [1,1,2,6] = (10) -- reroll 1x2
4d6 -> [4,5,6,6] = (21)
4d6 -> [2,1,3,1] = (7) -- reroll 1x2
4d6 -> [3,3,3,2] = (11)
4d6 -> [6,1,4,3] = (14) -- reroll 1
Rerolls:
http://invisiblecastle.com/find.py?id=973662
1d6 -> [4] = (4)
1d6 -> [2] = (2)
1d6 -> [4] = (4)
1d6 -> [2] = (2)
1d6 -> [4] = (4)
1d6 -> [6] = (6)
Applying those rolls in order, leaves:
4d6 -> [6,6,6,5] = (23) -> 18
4d6 -> [4,3,2,4] = (10) -- reroll 1 (4) -> 11
4d6 -> [2,4,2,6] = (10) -- reroll 1x2 (2 & 4) -> 12
4d6 -> [4,5,6,6] = (21) -> 17
4d6 -> [2,2,3,4] = (7) -- reroll 1x2 (2 & 4) -> 9
4d6 -> [3,3,3,2] = (11) -> 9 [dropped]
4d6 -> [6,6,4,3] = (14) -- reroll 1 (6) ->16
Total: 83
I didn't really do myself any favors there... proving, once again, that invisible castle hates me.