We say, something can go to infinity such as $\mathrm{e}^n$ goes to infinity faster than, say, $1+\frac{1}{2}+\frac{1}{3}+\cdots$. But they are all infinity.
Seems like we only care about the speed they go to infinity, not never think about what they looks like when they are already infinity.
For example, $(2+\sqrt{2})^n$ is an irrational number for any finite $n$. But when $n \rightarrow \infty$, $(2+\sqrt{2})^n \rightarrow $ integer. It not converges to any perticular integer number, but the "limit" belongs to the category of "integer".
$$(2+\sqrt{2})^n=2^n+C_{n}^{1}{2^{n-1}\sqrt{2}}+C_{n}^{2}{2^{n-2}{\sqrt{2}}^2}+\cdots$$
All odd items are integers, let the sum of all odd items $A$; and even items are irrational numbers, let the sum $B$. so
$$(2+\sqrt{2})^n=A+B$$
Noticing that $(2-\sqrt{2})^n=A-B \rightarrow 0$, we can easily find that $A>B$，and most importantly $B \rightarrow A$. A big irrational number $B$, is more and more close to integer $A$.
$(2+\sqrt{2})^n$ is infinitely close to integer.
Is there any branch of math research the property of infinity itself? An infinite integer has infinite decimal numbers, what's the difference with the irrational number BTW, which also has infinite decimal numbers.
The easiest way of formalizing this is in terms of the ultralimit popularized by Terry Tao. Here the equivalence class of a sequence becomes an integer in an extended number system called the hyperintegers or more generally the hyperreals. Thus, the ultralimit of the sequence of irrational numbers you constructed will become an infinite hyperreal infinitely close to the hyperinteger obtained as the limit of $A$'s. So the specific answer to your question: "Is there any branch of math research the property of infinity itself?" is affirmative. The branch in question is Abraham Robinson's framework for analysis with infinitesimals. See this popular calculus textbook for an elementary explanation.