Looking at the current Limited Enemies wiki page, it doesn't seem go into any detail about the drop rates, other than the fact that successive victories will increase the drop rate. A cursory search through the blogs yielded no results as well. I also saw a post on a different forum attempting to pinpoint the average number of victories, so I was inspired to investigate it myself, albeit more mathematically. If anyone wants to incorporate this information into the Limited Enemies page, that would be cool, since I don't know if my writing would fit stylistically.

First, I'll go over the drop rates for the different difficulties:

Difficulty (cats) Base Drop Rate Bonus Drop Rate
1 .39% +.50%
2 .46% +.60%
3 .56% +.80%
4 .71% +.90%
5 .99% +1.20%

So your chance to get the Limited Enemy drop is the base drop rate for the difficulty you are fighting plus the bonus drop rate for each one you have defeated until the bonus expires or you manage to get the drop. In the former case it resets, and in the latter it also resets but you keep the bonus of the last one you fought.

Anyway, if you are unlucky, the drop rate will keep increasing as you defeat more and more Limited Enemies, and if you are luck cursed (or luck blessed; it's all perspective), you could potentailly see it hit 100%, in which case you are of course guaranteed the drop. This mechanic is very reminiscent of pseudo-random distrubution, although the associated theoretical probability is not given.

So enough rambling, here are the results of my calculations. N = number of victories for a drop:

Note: The average is derived from the average drop rates, not the calculated probabilities
Difficulty (cats) Theoretical Probabilty Max N Most Probable N Average N
1 5.68% 201 15 17.61
2 6.22% 167 13 16.08
3 7.15% 126 11 & 12 13.98
4 7.64% 112 11 13.09
5 8.99% 84 9 11.28
Average 7.19% 126 11 & 12 13.90

To get these numbers, I basically recursively calculated the probability of encountering the Nth Limited Enemy without receiving the drop. That probability multiplied by the active drop rate gives you the actual drop rate for each individual N, and from there the most probable N and average N can be derived.

So what does this all mean? Assuming you face the average difficulty (all are encountered equally), then in the long run, the number of victories for the drop will be about 14. However, you are most likely to actually receive the drop on the 11th or 12th victory. In the worst case scenario of only encountering the 1 cat difficulty, the average will be about 18, and most are expected at the 15th victory. Contrarily, if you only encounter the 5 cat difficulty, the average will be about 11, and  most are expected at the 9th victory. Note that the most probable N is only most probable by a small margin, so think of it as a sweet spot, where it and surrounding numbers are all reasonable given moderate luck. Here's a supplemental (but truncated) graph depicting the likelihood of receiving the drop at a given number of victories:


If you aren't keeping track of how many you've defeated, for the average difficulty, 11 victories should be about 8.6% Bonus Drop Rate, 12 should be about 9.4%, and 14 should about 11%.

TL;DR: Look at the chart and graph above. Should be simple enough.

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