Topological Data Analysis (TDA) refers to an approach using concepts from mathematical topology to analyze datasets. TDA combines algebraic topology and other mathematics, mainly probability and statistics, to study the ``shape" of high-dimensional, incomplete, and noisy data. Using a basic quantifier in algebraic topology known as a Betti number, we investigate the topological features of extreme sample clouds generated by a heavy tail distribution on $\mathbb R^d$ with a regularly varying tail. We observe a certain layered structure consisting of a collection of ``rings" around the origin with each ring containing extreme random points which exhibit different topological behaviors. In particular, the growth rate of the Betti numbers and the properties of limiting processes all depend on how far away the region of interest in $\mathbb R^d$ is from the weak core - the area in which random points are placed densely enough to connect with one another. If the region of interest gets close enough to the weak core, the limiting process constitutes a new class of Gaussian processes. *This talk assumes no prior knowledge in algebraic topology.