Is homology finitely generated as an algebra? If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; free as an algebra; generated in positive degree? generated in negative degree?
What about commutative algebras or Lie algebras? (is anything else sensible to ask?) What if we reduce the grading from $\mathbb Z$ to $\mathbb Z/2$ (ie, differential super-algebras)?
What I'm really interested in is the case of $\mathbb Q$-DGLAs freely generated in homologically positive degree, because that corresponds to homotopy groups of finite complexes, but the commutative case is probably more familiar. I expect the answers to be the same, heuristically from Koszul duality.
 A: Another counterexample: let $A$ be the algebra $\mathbb{Q}[y,z]/(y^2) \otimes \bigwedge(x)$ with $x$ in degree 1, $y$ and $z$ in degree 2. Put a differential on this by $z \mapsto xy$. This is a commutative dga in characteristic 0 generated in positive degrees, but of course it's not free. Its homology is spanned by the classes $xz^i$ and $yz^i$ for all $i \geq 0$, and the product on the homology algebra is trivial. So it is infinitely generated as an algebra.
A: In the setting of cdgas, this is not true: that is, there is a simple example of a finitely presented coconnective cdga over the rational numbers whose homotopy groups are not finitely generated as an algebra. 
Namely, consider the cdga $R$ of (derived) global sections of the structure sheaf ("functions") on the punctured affine plane. Then $\pi_0 R = \mathbb{Q}[x,y]$ and $\pi_{-1} R  = \mathbb{Q}[x,y]/(x^\infty, y^\infty)$ where the latter refers to the cokernel of the map $\mathbb{Q}[x^{\pm}, y] \oplus \mathbb{Q}[x, y^{\pm}] \to \mathbb{Q}[x^{\pm}, y^{\pm}]$. (This is easy to check from choosing the standard cover of the punctured affine plane by the complements of the $x$ and $y$ axes, respectively.) 
Clearly, the homotopy groups of $R$ are not finitely generated as an algebra, but I claim that $R$ is finitely presented as a cdga. There is in fact an explicit finite presentation of $R$ due to Bhatt and Halpern-Leinster. Namely, let $M$ be the $\mathbb{Q}[x,y]$-module given by $\mathbb{Q}[x,y]/(x,y)$ and consider the natural map $\mathbb{Q}[x,y] \to M$ and its dual $\phi: DM \to \mathbb{Q}[x,y]$, where $D$ denotes Spanier-Whitehead duality in the (derived) category of $\mathbb{Q}[x,y]$-modules. Then the presentation of $R$ is that it is the homotopy pushout of the symmetric algebra of $M$, mapping in two different ways to $\mathbb{Q}[x,y]$, once via the map extending $\phi$ and once via the map extending zero. (As a $\mathbb{Q}[x,y]$-algebra, $R$ has the following universal property: to give a map from $R$ to some other $\mathbb{Q}[x,y]$-algebra $R'$ amounts to the condition that $R'/(x,y)$ should be contractible.)
