Near Misses
A near miss is a polyhedron that's almost a Johnson solid. Without formalizing what that means, we might ask that one should be able to construct the polyhedron out of paper (or some construction toy), using only regular faces, without noticing the discrepancy.
To my knowledge, there has never been a rigorous study of near misses, though Norman Johnson did encounter (and reject!) a number of them in his classication of Johnson solids. Evelyn Lamb wrote a fun article on this subject in 2017. The hendecagonal nearmiss at the bottom of the page ultimately played an important role in describing the structure of a protein cage in a Nature article, and spurred Agnieszka Kowalczyk to pursue some research on finding nearmiss polyhedral cages, which are related to solids but somewhat more flexible. I occasionally talk about new nearmisses on my blog.
Click on the image for a larger version.
O(*,3,*,[2]): 8 enneagons, 6 squares, 24 almost equilateral triangles.
The notation given is from the 2001 Symmetrohedra paper. 

I(1,2,*,[2]): 12 pentagons, 20 hexagons, 60 almost equilateral triangles.  
I(2,*,3,e): 12 decagons, 30 hexagons, 20 equilateral triangles, 60 almost squares.  
I(2,4,*,e): 12 decagons, 20 dodecagons, 60 almost
equilateral triangles (these are actually trapezoids with very
narrow tops), 30 very narrow rectangles.
This solid is very similar to a truncated buckyball. One imagines that by fiddling with it a bit, one could sacrifice the perfect regularity of the decagons and dodecagons in exchange for eliminating the narrow rectangular faces, resulting in a near miss. 

[no notation]
A solid constructed by inscribing a regular 11gon in every face of a pentagonal icositetrahedron. The result is a little loose, but there's room for improvement by starting with a differentlyparameterized base solid. Alex Doskey was kind enough to contribute WRL files for this solid: Solid, Leonardostyle. 