I want to consider a mixture of uniform distributions between $0$ and $t-U$ such that the $U$ is itself drawn from some mixing distribution. I want to pick a distribution such that the PDF (and ideally the CDF as well) of the resulting mixture has a nice closed form. I tried the exponential distribution, but ended up with the exponential integral which is non-elementary. The form of the mixed PDF is going to be (where $g_U(u)$ is the mixing distribution):
$$f_A(a) = \int\limits_{u=t-a}^t \frac{1}{t-u} g_U(u)du$$
Changing variables to $S:=t-U$ and assuming $0<U<t$, the mixture will be $$f_A(a)=\int s^{-1}\pmb{1}_{0<a<s}\,g_S(s)du=\int_a^ts^{-1}g_S(s)ds.$$ The integral will have an especially simple closed form if $S\sim tX$, where $X$ has a Beta distribution with parameters $\alpha>1$ and $\beta=1$: $$g_S(s) = g_X(s/t)(1/t)= (\alpha/t)(s/t)^{\alpha-1}\pmb{1}_{0<s<t}.$$ Then $$f_A(a)={\alpha\over \alpha-1}{1\over a}\left({a\over t}-\left({a\over t}\right)^\alpha\right)\pmb{1}_{0<a<t}.$$ (The scaled Beta distribution may have arbitrary parameter $\beta>0$, but in that case the mixture will involve an incomplete Beta function.)