Iterated Topological Hochschild Homology of ${\mathbb{F}}_p$ at odd primes

We compute homotopy groups of $n$-fold iterated topological Hochschild homology of ${\mathbb{F}}_p$ for $n\ge 1$ and odd primes. Identifying this as the Loday construction $T^n\otimes {\mathbb{F}}_p$, we construct a bar spectral sequence computing ${\pi }_{\ast }(T^n\otimes {\mathbb{F}}_p)$ and prove that this spectral sequence collapses at the $E^2$-page. Finally, we resolve the multiplicative extensions and determine the ring structure.