**Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? $\ldots$ Where can I read more about them?**

The paper

<blockquote>
Rogalski, D.; Sierra, S. J.; Stafford, J. T.,
Algebras in which every subalgebra is Noetherian. Proc. Amer. Math. Soc. 142 (2014), no. 9, 2983-2990. 
</blockquote>

introduces the term _supernoetherian_ for a not-necessarily-commutative $k$-algebra $A$ that has the property that all subalgebras of $A$ are both (i) finitely generated and (ii) Noetherian. In the commutative case, (ii) follows from (i) by the Hilbert Basis Theorem, so these are exactly the $k$-algebras asked about here. The authors of this paper do observe that, when $k$ is a field, the commutative supernoetherian algebras have Krull dimension at most $1$, but they do not classify them.