Group for which computing inverse is hard Is there any group in which not only discrete logarithm and finding square root are believed to be hard, but also finding the inverse of a given element?

Thank you.
Not to my knowledge. However, there are braid group and other nonabelian group based cryptosystems proposed where the difficult problem is the conjugacy problem in a nonabelian group $G$, i.e., given $g$ and $h$ in $G$, determine whether they are conjugate. This means finding an $x$ in $G$ such that $$g=xhx^{-1}.$$ 

Clearly inverses play a key part in these systems. You can search eprint.iacr.org for some papers. One notable paper in this field is the one below:

Cheon, J. H., &amp; Jun, B: A polynomial time algorithm for the braid Diffie-Hellman conjugacy problem, CRYPTO 2003.

None of these systems have been widely adopted, however. One of their attraction is resistance to quantum computing based attacks.