Take $A= \mathcal{C}([0,1])$ and $H$ the ideal of $A$ of function that vanish at $0$.
$H$ is a Hilbert $A$ module (as any ideal, with th nautral multiplication of $A$ and the scalar product $(x,y)=x^*y$ of $xy^*$ depending on if your talking of right or left modules) the inclusion of $H$ into $A$ is a continuous $A$ linear map and it has no adjoint.
Indeed an adjoint would be a map $p$ from $A$ to $H$ such that for all $y$ in $H$ and $x$ in $A$, $y^* p(x) = yx^*$ hence $p(1)$ would be an element of $H$ which is a unit for the ideal $H$, that does not exists.