People can, and do, put up full size 160m antennas. Usually "full size" means a 1/2 wavelength dipole, or a 1/4 wave vertical. The vertical works because the 1/4 element is "mirrored" by the ground plane, creating an image antenna, which makes it very much like a 1/2 wave dipole.
1/2 wavelength seems like a magic number because it is the minimum length required to attain resonance. Antennas can be made smaller and resonated by introducing additional, non-radiating reactive components like inductors or capacitors. However, shortening the antenna in this way makes the antenna less efficient.
Making the antenna longer than 1/2 wavelength isn't frequently done because in most cases there isn't much point to it. As the antenna first becomes longer than 1/2 wavelength, it stops being resonant. If made longer to 1.5 wavelengths, it becomes resonant again. However, it isn't any more efficient than a 1/2 wavelength antenna. In fact, it's likely less efficient, because the current must now travel over more antenna, which has more resistance.
What making the antenna longer does do is alter the radiation pattern. For example, compare a 1/2 wave dipole (in free space):
to a 1.5 wave dipole:
These are from antenna-theory.com.
Is this really useful? Usually no. One possible exception might be for verticals, where making the antenna longer (to a point, usually 5/8 wavelength) lowers the takeoff angle a bit.