(Per the asker's request, I'll be including romaji in this answer following Japanese text where it is used.)
As Earthling noted, they're finding ways to re-use both versions of the phrase in the sentence. So we started with:
Arumi-kan no ue ni aru mikan.
It uses the same 5 morae (a ru mi ka n), but changes where the split falls (in this case, before or after "mi"). In this case, it's also reasonably straightforward to interpret: An orange on top of an aluminum can.
Arumi-kan no ue ni aru mikan no ue ni...aru mikan!
In this case, stacking a second orange on top and getting it to stay allowed for adding one more use of it. Since み (mi) is also the Japense number for 3 (三つ mittsu), it's suggesting 3 objects, which further strengthens the pun in her favor.
Arumi-kan no ue ni aru mikan no ue ni aru mikan no ue ni aru...mikan!
This one gets a shade tricky. By putting another aluminum can on the stack he's actually managed to get it in there four times. But the way the breaks turns out a bit weird, since on the last one it should break after "mi" for the can, but the girl jumped in to ensure it sounded like another orange was added instead.
Hope that helps cover what you were looking for!