For each $k\geq 0$ we consider the Toeplitz algebra $\mathcal{T}_\lambda(M_k)$ of the monoid $M_k$ generated by the first $k+1$ generators in the infinite presentation of the Thompson group $F$. In this talk we will discuss some structural properties, generating sets and presentation. We construct a C*-correspondence whose associated Toeplitz algebra is canonically isomorphic to $\mathcal{T}_\lambda(M_k)$ while its Cuntz--Pimsner algebra is isomorphic to the boundary quotient $\partial\mathcal{T}_\lambda(M_k)$. We use this description to study nuclearity of these algebras. Since $M_1$ is the free monoid on two generators, $\mathcal{T}_\lambda(M_1)$ is known to be nuclear by results of Cuntz; $M_2$ is weakly quasi-lattice ordered and nuclearity of $\mathcal{T}_\lambda(M_2)$ follows from work of an Huef, Nucinkis, Sehnem and Yang. We establish nuclearity of $\mathcal{T}_\lambda(M_3)$ and discuss our approach to the remaining cases. This talk is based on joint work with A. an Huef, M. Laca, B. Nucinkis and I. Raeburn.