Not about tetration but what do you think about this paper?

[Ansus]

What also is interesting, is if we want a closure undder operation of definite integration of real-valued, but unbounded functions, in such a way that it preserves order (that is, if \(f(x)>g(x)\), then \(\int_a^b f(x)dx > \int_a^b g(x)dx\)) we get the set of countable surreals. And if we put countable surreals under the integral, we get continuum-sized surreals. Thus, the both sets come naturally from the attempts to find a closure of reals under integration. And it seems, one cannot get even greater surreals this way (putting continuum-sized surreals under the integral will not produce anything even greater than continuum-sized).