How far are we from the edge of the Universe? Following the Big Bang the Universe continues to expand, presumably and roughly equally in all directions. It is understood that the Big Bang occurred 13.798 ± 0.037 billion years ago.

Is there any way for us to know how far we are away from the nearest edge of the expansion front of the Big Bang?
In the standard models of the Big Bang, there is no such thing as the "edge of the expansion front". The universe, as far as we can see and as far as standard cosmology assumes, is homogeneous and isotropic on the large scale, so there is no edge or anything analogous to an the shock front of an explosion.

According to the seven-year WMAP results [pdf], the proper distance to the surface of last scattering that emitted the cosmic microwave backgrounds is approximately $46.0\,\mathrm{Gly}$, which is basically as far as we can actually see. The true cosmological horizon slightly more distant that this, closer to $46.6\,\mathrm{Gly}$. What's beyond the horizon is not known.

Another cosmologically significant distance scale is the Hubble radius, at which the galaxies comoving with the Hubble flow recede from us at the speed of light. It is $c/H_0 = 13.9\pm0.3\,\mathrm{Gly}$.