Except that huge inequalities exist in socialistic economies as well, and capitalism means the middle class of today live as well or better than the richest people in the world did in the 1920s.
I even acknowledge that the poor are getting richer. The problem is just that they are getting richer at a slower pace than the rich, and unless this problem is addressed through taxation, then it will lead to economic stagnation or worse.
Competition is the best bar to too much competition.
Picketty's explanation is that capital's reward is higher than work's, but I'm not bent on defending that explanation (even if until a better one comes up, I'll stick to Picketty's).
As far as I know there is no counter example of the phenomenon itself (whatever the underlying reason is).
Even statistical anomalies like Bill Gates or Steve Jobs got more money, once they became slightly rich, from the capital they owned than from the work they produced.