This isn't really a question rather than my thoughts on these things; I initially had questions but believe I managed to answer them. Regardless, here goes. Feel free to correct any mistakes, as I'm a bit new to this whole thing. For the moment, I'll only be considering systems using classical logic.
This idea initially occurred to me while thinking about trying to construct surreal numbers in New Foundations and other theories with unrestricted (albeit odd) comprehension. In trying to construct surreal numbers in naive set theory, we have the Burali-Forti-like problem that if S denotes the set of all surreal numbers, {S|} is greater than any surreal number; in particular, {S|}>{S|}. This brings up the question of whether any theories such as Quine's New Foundations or the positive set theory $GKP^+_\infty$ admit a similar paradox.
First, I'll give a treatment of the problem in Ackermann set theory with the axiom of foundation (which is needed to define surreal numbers via transfinite recursion.) If we consider the class S={x | x is a surreal number}, the paradox above yields that S is a proper class; an interesting thing to note is that this means there must be proper class surreals (as otherwise S would itself be a set.) Nothing unusual here; we just have the situation of ordinals all over again.
Now to ask the question in set theories admitting unrestricted comprehension. To investigate this question, we need a formula which is true for surreal numbers and false for any other sets, or equivalently we need to ask the question of exactly what a surreal number is. Ordinarily we define them via transfinite recursion on the ordinal rank of a set, which requires the axiom of foundation to work; however, the axiom of foundation doesn't hold in these theories. We then define surreal numbers as (Quine-Rosser) ordered pairs (x,y) such that x and y are both hereditary sets, every element of x is a surreal number, every element of y is a surreal number, and every element of x is smaller than every element of y. If, after fully formalizing this condition, we get that S is a set, it cannot be hereditary; otherwise (S,∅)∈S, which (I think) contradicts our assumption that it is hereditary. Thus (S,∅) is not a surreal number as we need to obtain a paradox.
I haven't explored this problem much further, in particular it may be possible to define "surreal number" in a manner which doesn't require the ordered pair to have hereditary components. A question nagging at me is what happens when we try to do this in double extension set theory.