Non-trivial approximate subrings of ${\bf R}$ or of ${\bf F}_p$.

The existence of such objects is ruled out by a number of "sum-product theorems", a typical one of which asserts that given a subset $A$ of ${\bf F}_p$ that is not extremely large or extremely small, either the sum set $A+A$ or the product set $A \cdot A$ has to be significantly larger than $A$.

On the other hand, one can improve upon the "trivial bound" in many arguments in arithmetic combinatorics or combinatorial geometry by analysing a putative configuration that attains this trivial bound and showing that it ultimately must arise from an approximate subring. Some early examples of this are in

Bourgain, Jean; Katz, N.; Tao, Terence C., A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14, No. 1, 27-57 (2004). ZBL1145.11306.

There are now dozens of other places where this sort of argument appears. (Analogous arguments also appear in other fields, e.g. using the "group configuration theorem" from model theory, or the "group chunk theorem" in algebraic geometry.)