PSR J1906+0746 is a young pulsar in the relativistic binary with the second-shortest known orbital period, of 3.98 hr. We here present a timing study based on five years of observations, conducted with the five largest radio telescopes in the world, aimed at determining the companion nature. Through the measurement of three post-Keplerian orbital parameters, we find the pulsar mass to be 1.291(11) M sub(middot in circle), and the companion mass 1.322(11)M sub(middot in circle), respectively. These masses fit well in the observed collection of double neutron stars (DNSs), but are also compatible with other systems where a young pulsar such as J1906+0746 is orbited by a white dwarf (WD). Neither radio pulsations nor dispersion-inducing outflows that could have further established the companion nature were detected. We derive an Hi-absorption distance, which indicates that an optical confirmation of a WD companion is very challenging. The pulsar is fading fast due to geodetic precession, limiting future timing improvements. We conclude that the young pulsar J1906+0746 is likely part of a DNS, or is otherwise orbited by an older WD, in an exotic system formed through two stages of mass transfer.