Short answer: Yes and yes. For the first question, you could take any $EXPTIME$-complete problem. For the second you could take any $NEXPTIME$-complete problem.
Your first question is answered by the problem:
Given a deterministic Turing machine M, string x, and integer k in binary, does M accept x within k steps?
The output is one bit (yes or no). The above problem is $EXPTIME$-complete, hence it requires time that is exponential in the lengths of M, x, and k. It is crucial that k be written in binary. If it were written in unary (as a string of k ones) then it is solvable in polynomial time by direct simulation.
Your second question is answered by the problem:
Given a nondeterministic Turing machine N, string x, and integer k in binary, is there an accepting computation path in N(x) that has at most k steps?
Again the output is just one bit. The above problem is $NEXPTIME$-complete, and hence requires exponential time even on a nondeterministic machine. Again it is crucial that k is written in binary; if it were written in unary then the above problem is $NP$-complete, and is more commonly known as the "Bounded Halting Problem".
This can all be found in the early chapters of any text on complexity theory.