  We have used the Keck 10m telescope to count objects as a function of image
size in 2 high galactic latitude fields covering 1.5 arcmin^2 and reaching 50%
completeness at K=24 and J=24.5 for stellar sources. Counts extend ~1 mag
deeper in K than surveys with other telescopes; complement Keck surveys
providing counts at comparable or shallower depths but not utilizing image
structure; and extend by several magnitudes the J band counts from other
surveys. We find the surface-density of objects at K=23 to be higher than
previously found (~500,000/mag/deg^2), but at K<22 to be consistent with most
other surveys in amplitude and slope (~0.36). J band counts have similar slope.
J and K counts are in excess of our empirical no-evolution models for an open
universe, and a factor of 2 higher than mild-evolution models at J and K ~ 23.
The slope of the model counts is insensitive to geometry even in the
near-infrared because the counts are dominated by low-luminosity (<0.1L*)
objects at modest redshift (z<1) with small apparent sizes (r05<0.4", i.e. <4
kpc). The observed counts rise most steeply for these smaller objects, which
dominate fainter than K=22.3 and J=23.3. However, the greatest excess relative
to no-evolution models occurs for the apparently larger objects (median
J-K~1.5). The size and colors of such objects correspond equally well to
luminous (>0.1L*), galaxies at 1<z<4, or progressively more diffuse,
low-luminosity (0.001-0.1L*) galaxies at z<1. We rule out the excess is from
very low luminosity (<0.0001L*) red galaxies at z<0.25. There is a deficit of
galaxies with red J-K colors corresponding to luminous, early-type galaxies at
1<z<3. Assuming the deficit is due to their appearance as blue galaxies, they
account only for 10-30% of the excess of large, blue galaxies. [abridged]
