Water expands when it heats up. If you heat water in a container that prevents it from expanding, will its temperature top out -- maybe around the boiling point?
And if not, will it still turn to gas without room to expand?
No, its temperature will not stop rising. Substances have both a temperature, and a density. Remember in chemistry you had two different heat capacities, constant pressure, and constant volume. The water will still heat up, although the heat capacity is the constant volume heat capacity. At some temperature, it becomes a gas, with the same density as it started with.
In every day use of physics, you are talking of a pressure cooker situation: a strong container and heat applied. The pressure cooker has a valve that lets off steam so as to stop the temperature at a point, higher than the boiling point of water at normal temperatures and pressures. That is the whole point of it, to cook at higher temperatures faster.
Study the phases of water in this article.
When you apply heat to a closed container with water, you change its phase into steam and mixed, raising the pressure. In the end all the energy goes to heat because there is no other outlet, and energy is conserved.
In the first plot you see that when the pressure rises, temperature rises, and even though it is a many valued function, i.e. the physical path may take different values depending on the initial conditions, still the answer is that the temperature will keep on rising until the container bursts from the pressure build up.