I know that the boundary of a single $d-$dimensional simplex would be a chain of $d+1$, $d-1$-dimensional faces. However, is it possible for a single $d-1$-dimensional simplex to be a boundary of a chain of $d-$dimensional simplex. Intuitively, it shouldn't be the case. However, I can't come up with a counter example or proof.
Any ideas?
Are you talking about singular simplices or simplices in a simplicial complex? If the latter, use the fact that $\partial^2=0$, so the boundary of any chain must itself have zero boundary. However the boundary of a single simplex is nonzero, so it cannot itself be a boundary.
It is possible for a single simplex to be a boundary in the singular case.