Helium superfluid-3 has got at around 2K a special liquid state with 0 viscosity with interesting properties. I am really interested in what would happen at 0K. With no kinetic energy in the particles of the compound, would it be solid? Do elements with no solid state at all exist?
First of all, we cannot achieve temperatures of 0 kelvin in experiments, that is forbidden by quantum thermodynamics.
Currently, experiments can go down to seriously low temperatures of nanokelvins (source: a friend is working in that field, but spontaneously the only reference I can give you is Wikipedia.)
At those temperatures the phases of 'solid' or 'liquid' don't really exist anymore. Rather collective quantum effects play a role, that make new phases appear, namely Bose-Einstein condensates and Fermi gases. What phase appears depends on the particles spin, which allows or disallows them to occupy the lowermost energy level.
Those exotic phases of matter count however still as 'condensed matter', which all elements at cold temperatures will have. The classical solid phase, which you're asking about, can be circumvented this way, but not at all pressures, as others correctly pointed out.