In the textbook treatment the pressure decreases exponentially with height as $P(z)=P(0)\exp(-z/H)$ where $H=kT/mg$ is the scale height of the atmosphere. This assumes that the gravity $g$ is constant with altitude, which is a decent approximation for planets with an atmospheric height much smaller than their radius. Since $H$ is about 8 km for Earth, this makes sense to use.
For an atmosphere with Earth-like temperature $T$ and mean molecular mass $m$ it would hence seem that for the tungsten planet that if we want $P(0)$ to be like on Earth $g$ must be equal to Earth's 9.82 m/s$^2$. So if we have a planet with $g=G\rho (4\pi r^3/3)/r^2$, we get $r=3g/4\pi G\rho$. For $\rho=19300$ kg/m$^3$ this gives $r=1820$ km.
The limit is met somewhere before white dwarf density. They have a density of about a billion kg/m$^3$ and a 35 meter radius white dwarf would have terrestrial surface gravity. At this point the above approximation has broken down (the scale height is bigger than the radius), and besides the low gravity would not be enough to hold it together. The real determinant of the limit will be the highest density material you can get that is stable at (near) zero pressure. For molecular matter tungsten is likely close to the limit, and we have reason to doubt degenerate white dwarf matter is up to the job.
Note that small but less dense bodies can have quite dense atmospheres. Titan has radius about 2550 km and a surface pressure 1.48 times Earth, despite lower surface gravity. The trick here is that the atmosphere is colder and slightly denser.