I didn't dive too deep into the bit-wise implementation details, but overall this looks very promising.

The first big issue I would have with Posits is the lack of a Not-a-Number (NaN) representation (as indicated on slide 16).

NaN values have the property of making any calculation they are used in also result in in a NaN.

I've found this property of NaNs to be invaluable in many contexts, especially as a no-data value regular data.

For example, this is useful when using bllinear filtering of floating point textures on a GPU -- any time a NaN texel is encountered, the entire fragment is discarded. In this manner, NaN values can be used to mask out a region without introducing a separate texture lookup.

I also love the simple, albeit obtuse, check for NaN:

  // Don't remove this if! It's checking if value is a NaN
  if (value != value) {
     // value is NaN...
  }