How small can a planet be and still have Earth-like gravity? A planet made from denser material than Earth might have equal gravity to Earth but a smaller radius. How small can a planet be and still have Earth gravity?

Ideally it should be habitable by humans, so not made of anything dangerous, radioactive or unstable.
The surface gravity of a planet is very close to 
$$g=\frac{4\pi G}{3}\rho r.$$
With $g$ to be kept constant, and $\frac{4\pi G}{3}$ a constant, we need
$\rho_Pr_P=\rho_Er_E$, or
$$r_P=\frac{\rho_E}{\rho_P}r_E,$$
with $\rho_E=5.515 \mbox{ g}/\mbox{cm}^3$ the mean density of Earth, $r_E=6371.0 \mbox{ km}$ the mean radius of Earth, $\rho_P=22.59\mbox{ g}/\mbox{cm}^3$ the density of densest natural element osmium, and $r_P$ the radius of the fictive osmium planet.

Hence $$r_P=\frac{5.515}{22.59}r_E=0.2441~r_E=1555\mbox{ km}.$$

Some compression of the core of an osmium planet due to pressure is neglected.