Firstly, i am not sure what part of the Questin is ambiguous.

However, i can say that you never reach the speed of light. NotLKH, we have assumed that putting balls in/out of the box takes no time at all.

I think the problem comes with treating the balls as objects, not every ball as '1'.

You see, if we treat every ball as '1', and adding/subtracting to box as +/-, then sure, the sum ends up as (10-1)+(10-1)+(10-1) etc. = 9+9+9... = infinity

However, the balls are objects, you can't take any 1, you have to take a certain 1.
I.e. the sum then becomes:
(11+12+13+14+15+16+17+18+19+110 - 11)
+ (111+112+113+114+115+116+117+118+119+120 - 12)

This is a telescoping sum, which cancels out to 0.
I.e. if the balls ar treated as a value, "1" then you get infinity. If you treat them as objects (i.e. strings: "ball1", "ball2" etc) then they cancel out to 0.

This is what i don't agree with in Guv's (10-1)+(10-1)+... case, as you are really taking an arbitrary "1", from a 10 which the 1 isn't in. Sounds weird, i know.