Discrete log accumulator without pairings? Here $g$ is some fixed generator of a discrete log group. I don't want the group to be bilinear for efficiency and BDH-skepticism reasons.

Is anyone aware of a discrete log accumulator? What I mean specifically is some function $f(x, A)\mapsto A'$ (that is, $A$ is the accumulator value; $f$ adds $x$ to the accumulator, changing its value to $A'$) such that given $(g^x, A')$ anyone can check whether $x$ was placed in the accumulator.

So I'm roughly asking for an accumulator with the property that an accumulated element $x$ has $g^x$ as a witness.
Sure.  Use the strong RSA assumption.  The accumulator of $x_1,\dots,x_k$ is $A = g^{x_1 x_2 \cdots x_k} \bmod n$, where $n$ is a RSA modulus and $g$ is a fixed base.  To prove that the accumulator $A$ contains $x$, exhibit a value $h$ such that $h^x=A \pmod n$.  This is secure under the strong RSA assumption, and has a discrete log "feel" to it.