I will discuss one instance of the so called Hastings-Levitov planar aggregation model, consisting of growing random clusters on the complex plane, built by iterated composition of random conformal maps. In 2012 Norris and Turner proved that in the case of i.i.d. maps the limiting shape of these clusters is a disc. In this talk I will show that the fluctuations around this asymptotic behaviour are given by a random holomorphic Gaussian field $F$ on $\{\left|z\right| > 1\}$, of which I will provide an explicit construction. The boundary values of $F$ perform a Gaussian Markov process on the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of the whole plane Gaussian Free Field to the unit circle.