We show that the strong-coupling physics inherent to the insulating Mott state in 2D leads to a jump in the chemical potential upon doping and the emergence of a pseudogap in the single-particle spectrum below a characteristic temperature. The pseudogap arises because any singly occupied site not immediately neighboring a hole experiences a maximum energy barrier for transport equal to t(2)/U, t the nearest-neighbor hopping integral and U the on-site repulsion. The resultant pseudogap cannot vanish before each lattice site, on average, has at least one hole as a near neighbor. The ubiquity of this effect in all doped Mott insulators suggests that the pseudogap in the cuprates has a simple origin.