There are at least two or three different ways to accomplish this, depending on what variations you count as different solutions.
At bottom:
Make 49 pairs (leaving 2 wires unpaired).
At top:
Find 2 unpaired wires and match the 49 pairs. Take 1st unpaired wire and label it #1. Take 2nd unpaired wire label it #2. Label each wire in a pair #3 & #4, next pair of wires #5 & #6 and so on until the last pair is labeled #99 & #100. Connect #3 to #2, #5 to #4, #7 to #6 ... and #99 to #98. (Connecting all wires in series except #1 [and #100 --pjt]). _p At bottom:
With [conductance --pjt] meter connected to a pair and one unpaired, determine which lead is #1 (the only lead not connected in series, if meter is zero then it is #1, if infinity than it is #2). The other unpaired is then either #1 or #2. Disconnect 49 pairs but keep the wires togerther as mates. Connect meter to #2 and find #3 (the meter will read infinite). Mark #3's mate as #4 and reconnect #3 to #4. Leaving meter connected to #2 find #5 (the meter will read infinite). Mark #5's mate as #6 and reconnect #5 to #6. Continue in this fashion until you find and label #99, then the last wire will be #100.
At bottom:
Connect wires to groups of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2 and 1 wires.
At top:
Identify the groups. Label each wire based on its group: (A11), (A21,A22), (A31,A32,A33), (A41,A42,A43,A44), ... (...,A99), (A01,A01...A09,A00), (B99,B98...B91) ... B11.
Then take the A group with 10 wires in it. Take one of its wires and connect it to a 9 wire group (e.g. A09 to A99), take another an connect it to a 8 wire group (e.g. A08 to A88), et cetera, until you connect one to the single A wire (A01 to A11) and the tenth one (A00) remains unconnected.
Which of each pair of wire groups with the same number of wires is labeled A and which is B is unimportant, the difference can be seen later because one of the A group wires is connected to a 10 wire group but none of the B wires is.
Do the same detachment for each of the wires in group A9x and iterate for A8x ... and then for each B group.
At bottom:
Disconnect (electrically) the groups made at bottom but keep the wires together. Now it's trivial to check which wire is which.
The idea that I invented, before hearing any of the above, is somewhat similar to Harri's solution.
At bottom:
Make groups of 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13 and 14 wires.This totals 99 wires, and you have 1 unconnected. Climb up. (Ouch.)
At top:
Measure the wires. Now you know the one distinct wire which is not connected, and you know which wires belong to the groups of 2, 3, 4, 6 ... 14 wires.
Take one wire from each of the groups and connect it to the known distinct wire (which was originally unconnected). This gives you a group of 13 wires, and you have 12 remaining groups, with 1, 2, 3, 5, 6 ... 13 wires.
Take, again, one wire from each of the groups of 2, 3, 5, 6 ... 13 wires, and connect it to the one lonely wire (which is the remaining one of the original 2 wire group). This gives you a group of 12 wires, and you have 11 remaining groups, with 1, 2, 4, 5, 6 ... 12 wires.
Take one wire from each of the 2, 4, 5, 6 ... 12 wires, and connect it to the one lonely wire (which is the remaining one of the original 3 wire group). This gives you a group of 11 wires, and you have 10 remaining groups, with 1, 3, 4, 5 ... 11 wires.
Take one wire from each of the 3, 4, 5 ... 11 wires, and connect it to the one lonely wire (which is the remaining one of the original 4 wire group). This gives you a group of 10 wires, and you have 9 remaining groups, with 2, 3, 4 .. 10 wires.
Now take one wire from each group, and just leave it unconnected so far. You get 9 distinct wires, and you have 9 groups of 1, 2, 3 ... 9 wires. Mark the unconnected wire that was one of the 2 wire group (of originally 6 wires).
Take one wire from each of the groups of 2, 3, 4 ... 9 wires, and connect it to the one lonely wire (which is the remaining one of the original 6 wire group). This gives you a group of 9 wires, and you have 8 remaining groups, with 1, 2, 3 ... 8 wires.
Continue this iteration, producing groups of 8, 7, 6, 5, 4, 3 and 2 wires. You have 1 wire left, and you cannot leave it unconnected because then you would have 2 unconnected wires left of the original 14 wire group. Therefore, connect it to the other unconnected wire that you marked before (of the original 6 wire group). You have now 8 unconnected wires, two groups of 2 wires, and groups of 3, 4, 5, 6, 7, 8, 9, 11, 12 and 13 wires. (This totals 8+2+2+3+4+5+6+7+8+9+10+11+12+13 = 100, so it matches.) Climb down. (Phew!)
At bottom:
Measure the wires to find out the new groupings made at top floor. By cross-checking the groups, you can now identify which wire is which. The only problem is that there are two groups of 2 wires; these you can identify because one of them has wires from original groups of 13 and 14, and the other has wires from the original groups of 6 and 14.
Approvable answers were received from the following persons: