There’s no analyzing this construction.
The construction in the title seems to be an impersonal(-like) or passive(-like) construction which might resemble, to some extent, a middle(-like) Icelandic construction with -andi participles. The meaning is similar to This construction is unanalyzable. That is, there is some “ability” modality in there somewhere. And it seems to require a polarity marker, which is usually negative:
1) There’s no analyzing this. ‘One can’t analyze this’
2) * There’s analyzing this. intended: ‘One can analyze this’
But the polarity marker can be contrastive. The underlined portion of the following example, found online, has the contrastive/emphatic polarity marked with too and is acceptable to me:
3) A: There is no denying that the writers seem to have taken two or three steps backwards, rather than continuing where they left off at the GF.
B: There is too denying it! Them taking a break from the shipping for an episode doesn’t mean that they’ve taken a step backwards, it means that they’re taking a pause.
For me, I think affirmatives are possible with contrastive focus on the copula, as in I disagree; I think that there most certainly is denying the usefulness of that research program.
Finally, even though I don’t accept affirmative examples, I think I can say something like There’s no analyzing this construction… or is there? Tags also seem fine, as in There’s just no talking to him, is there?, while simple yes/no questions are unacceptable, as in *What’s John like? Is there talking to him?
Even more interestingly, there are examples with a by-phrase; though this might be marginal in many cases, it is acceptable at least with exceptives, as in the example in (4) (also found online, and acceptable to me):
4) There is no denying Zardari’s stellar behavior and performance, not even by his critics and naysayers.