Interesting question, here is what I think :
There are 2 possibilities regarding consciousness ( the I ), since we do not know how does the brain produce my conscious experience :
A- Either my consciousness is repeatable (if you build a similar brain)
B- Or it is very unique, such that even if you rebuild my brain, the new 'me' would only be a twin with the same personality.
Let's adopt a 100% agnostic position, so A 50% and B is 50% probable.
We are only concerned with A, so let's assume A is the case. Then we will have two other possibilities regarding existence itself (one or multiple Universes):
C- Either either existence is infinite in time (so that it has no beginning, or no end, or neither).
D- Or, existence has a beginning and end.
If we adopt a 100% agnostic position (since even if our Universe "started" with a big bang, 'started' here with cross-fingers. We do not know whether there is a Multiverse or no, so we do not know pretty much everything about whether existence has beginning or not) ... then, absolute agnosticism tells us that C is 50% probable and D is also 50% probable.
We are only concerned with C ... so let's ignore D.
We have also another condition, the Universe must NOT include an infinite amount of possible configurations, that is, different configurations and arrangement of the matter in all existence has to be finite. Because if it is infinite, then the probability of there "me" happening again decreases to, practically, zero (becomes very improbable in other words).
E - Possible configurations of matter in all existence is finite.
F - Possible configurations of matter in all existence is infinite.
I personally cannot adopt a very agnostic view (like 50-50%) here, because I know that the Universe is probably infinite in space based on modern physics measurements, so our Universe is probably flat, which implies infinite in space, which highly suggests that the amount of stuff in the Universe is probably infinite, let's give it a 66% probability (the Universe can still be finite and spherical and have a very small curvature that cannot be detected, if it is too large).
So, E gets a 33% probability .
But why exactly did I give it 66% probability?
Because there are two types of infinite Universes (in space) : Flat and saddle shaped Universe, while there is only one finite Universe (spherical Universe). so 66% to 33%. That's why I am not fully agnostic about this one.
Now, for there to be an eternal recurrence of the exact same me, A (P=50%) and C(P=50%) and E(P=33%) must all be true , and probabilities here are with respect to the maximum agnosticism I can express.
the probability of the 3 being true is : 0.5 * 0.5 * 0.33 = 8.25%
So, probably eternal recurrence is false (even if you are 100% agnostic about the probabilities), let's see .. if you are 100% agnostic about E then it becomes 50% probable.
Thus, 0.5^3 = 12.5%
So, the probability of eternal recurrence is between 8.25 and 12.5%
Because in my opinion I think that these are the ingredients :
1- Existence must be eternal.
2- Consciousness must be repeatable
3- Possible configurations must be finite.
That's it ^^, we don't know. But the idea is both scaring and comforting , why?
1- It is scary because it means that I will experience all possible lives (not only the same life repeated over and over, but also different lives), which implies that I will experience all physically and mentally possible sufferings.
2- It is comforting, because it implies that I will live and experience all possible pleasures.