$L_1 =\{a^{2^n}\}=\{aa,aaaa,aaaaaaaa....\}$
It's Exponential power.
So CSL.
$L_2$ = $\left \{ (a^{n})^{m}.b^{n}|n,m\geq 1 \right \} =\{a^+.b \ |n=1,m \geq1\} \cup \ \{a^n.b^n |n\geq1,m=1\}\cup \ \{a^{2n}.b^n |n\geq1,m=2\}\cup \ \{a^{3n}.b^n |n\geq1,m=3\}...$
According to me, It looks like CFL.
But it's not CFL. It's CSL.
CFL is not closed under infinite union.
Of course, it's not DCFL so DPDA can't be sufficient.
As Union is infinite and No. of states in NPDA is finite so NPDA is also not sufficient for it.