Is there any resource that might help non-experts gains some understanding of why
the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$;
whether $\theta_j=\theta_6$ exists in the "$128$-stem"), i.e., why the celebrated Hill-Hopkins-Ravenel 
proof technique fails in this last remaining case?
I read a delightful exposition by Erica Klarreich ("Mathematicians solve 45-year-old Kevaire Invariant Puzzle,"
The Best Writing on Mathematics, 2010, 373ff, Simons Foundation link), which piqued my interest.
But Michael Hopkins' online presentation slides announcing the result
in 2009 ("Applications of algebra to a problem in topology") are beyond my ken.
Perhaps this an area too abstruse for all but the experts?
I'd appreciate pointers to expositions.  Thanks!
I'll give a shot at an answer.  The relevant dimensions are of the form $2^j-2$. For 
$j\leq 4$, it is easy and classical that we can construct manifolds of Kervaire invariant one. The
problem was ``reduced'' from differential topology to pure stable homotopy
theory by Browder in 1969.  Direct calculational methods in
homotopy theory were used by Barratt, Jones, and Mahowald to construct a
cell complex that can be used to solve the homotopy theory problem and prove 
that such manifolds also exist in dimensions 30 and 62.  I believe a construction 
of such a manifold has been worked out in dimension 30, but that has certainly not 
been done in dimension 62. Periodicity phenomena play a huge role in modern stable 
homotopy theory, and a crucial feature of the Hill, Hopkins, Ravenel proof is a 
periodicity of order $2^8 = 256$.  That enables them to solve the stable homotopy problem 
and prove there is no manifold of Kervaire invariant one for $j\geq 8$.  The reasons $j=7$ 
is so hard are several. Nobody has a really good reason for guessing which way the answer 
will go. There is no reason to expect a relevant periodicity of order $2^7$. Direct calculation
of the Adams spectral sequence through dimension $126$ is just plain hard: the
calculations blow up.  There is a chance that the methodology of Barratt, Jones,
and Mahowald might extend to prove existence (if that is how the answer turns out!),
but it will probably be much harder to prove nonexistence (if that is the answer).   