The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle systems, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what are the appropriate scaling of the peaks and valleys of the KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets.