Are there reactions like
$X(s) + n Y(l) \to X \centerdot Y_n(l)$
that are endothermic? What are X and Y then?
The material X should also be soluble to material Y.
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You will find the answer in probably any textbook on chemical thermodynamics, or in a thermochemistry chapter in most physical chemistry textbooks, or in Wikipedia's enthalpy of dissolution page:
So, it can have any sign. The linked page also gives data for common compounds in water, some of them endothermic, some of them exothermic.
Regarding why the substance is soluble why its dissolution is endothermic, you have to remember that the reaction takes place if $\Delta_r G$ is favourable (i.e. negative), and $\Delta_r G = \Delta_r H - T \Delta_r S$. Overall the free energy must be negative for dissolution to occur (on a thermodynamic basis; kinetics are another issue), not the enthalpy.