GrossOne? The arXiv blog's pick of the day. The arXiv blog having chosen GrossOne for its daily pick of today, I read the arXiv paper concerned, and posted a comment there. The arXiv blog used to be quite high profile as these things go.
Is there an existing well-known (but not to me) mathematical construction that does what Sergeyev proposes?
 A: This is a comment, but it is too long and complex to be added in the usual manner.
The proposed mathematical structure is not in the paper above, which references three other sources by the same author. I was able to find one online:
A new applied approach for executing computations with infinite and infinitesimal quantities (arXiv:1203.3132)
This paper apparently introduces the structure, but does not define it very formally.  From the paper:

[The Infinite Unit Axiom] is added to axioms for real numbers (remind that we consider axioms in sense of Postulate 2). Thus, it is postulated that associative and commutative properties of multiplication and addition, distributive property of multiplication over addition, existence of inverse elements with respect to addition and multiplication hold for grossone as for finite numbers

leaving to the reader to decide exactly what those are* and how to extend them to include this new element. It seems that we start with a complete ordered Archimedean field and adjoin a new element to result in, at least, a field. (I intentionally omit completeness and order, q.v. below.)
The Infinite Unit Axiom has three parts:

Infinity. Any finite natural number n is less than grossone

This limits the order on the new structure. By the Archimedean property, any real x is less than grossone.

Identity. The following relations link grossone to identity elements 0 and 1
  [six formulas: multiplication by 0, division by itself, and exponents work for grossone as for real numbers]

This supports the quote about grossone acting like finite numbers.

Divisibility. For any finite natural number n sets Nk,n, $1\le k\le n$, being the nth
  parts of the set, N, of natural numbers have the same number of elements indicated by
  the numeral grossone/n where $$\mathbb{N}_{k,n}=\{k,k+n,k+2n,\ldots\},\ 1\le k\le n,\ \bigcup_{k=1}^n\mathbb{N}_{k,n}=\mathbb{N}.$$

This gives some kind of definition of what cardinalities look like in this system.

Normally given a system like this I'd look for contradictions, but the definitions are too wishy-washy to nail that down easily.  My assumption that the "axioms for real numbers" were the ordered field axioms plus completeness is inadequate, for example, since apparently exponentiation is included as well.  Perhaps someone will find these notes useful, though, so I leave them here.
A: Copying over here what I posted there:

I haven't read the references the author cites, but I don't see a lot of merit in this paper. The final answers seems to be simply taking the formula for the area of the Sierpiniski carpet (respectively Menger sponge) after n steps replacing the symbol n by the GrossOne symbol.
Note that this is not assigning an actual area to the Sierpinsiki
  carpet! If we defined the Sierpinski carpet by a different procedure,
  we would get a different answer. For example, if we lumped together
  every two steps, we would get the answer that the area of the
  Sierpinski carpet after GrossOne steps is (8/9)^{2*GrossOne}. The
  point of measure theory (which assigns the Sierpinski carpet measure
  0) or more generally Hausdorff measure (which tells us that the
  Sierpinski carpet has (log 8/log 3)-dimensional area equal to 1) is to
  assign areas to subsets of the plane no matter how they are defined.
I also strongly disagree that mathematicians do not have good tools to
  describe behavior of functions close to infinity. For this sort of
  simple exponential decay, the ordinary language of asymptotic
  expansions does excellently. (See, for example, de Bruijn's Asymtotic
  Methods in Analysis.) For more complicated decay, the theory of
  transeries is excellent (see Edgar, Transseries for Beginners, Real
  Anal. Exchange 35 (2010), also available at
  http://www.math.osu.edu/~edgar.2/preprints/trans_begin/beginners.pdf )
  for a good introduction. And, for these sort of specific fractal
  examples, this is what Hausdorff dimension and Hausdorff measure were
  invented for! See any textbook on Fractal Geometry.
I would not expect this paper to appear in a quality journal.
I usually wouldn't write something like this in public. This is what
  I'd normally send to an editor who contacted me to referee something
  like this. (Also, I am not an expert on fractals or measure theory, so
  I am an unlikely choice of referee.) But I worry that the arXiv blog
  has as much visibility as all but the best journals, so it is a major
  problem if it singles out something like this. For example, your
  software tells me that 111 people are reading this right now. How many
  people are reading the Bulletin of the AMS right now?
May I ask what procedure the arXiv blog uses to vet the papers it
  recommends? I would hope that they are sent out to experts for quick
  opinions as to general merit (similar to the ones JAMS and other top
  math journals solicit before beginning the reviewing process).

