  We present a device-independent randomness expansion protocol, involving only
a constant number of non-signaling quantum devices, that achieves
\emph{infinite expansion}: starting with $m$ bits of uniform private
randomness, the protocol can produce an unbounded amount of certified
randomness that is $\exp(-\Omega(m^{1/3}))$-close to uniform and secure against
a quantum adversary. The only parameters which depend on the size of the input
are the soundness of the protocol and the security of the output (both are
inverse exponential in $m$). This settles a long-standing open problem in the
area of randomness expansion and device-independence.
  The analysis of our protocols involves overcoming fundamental challenges in
the study of \emph{adaptive} device-independent protocols. Our primary
technical contribution is the design and analysis of device-independent
protocols which are \emph{Input Secure}; that is, their output is guaranteed to
be secure against a quantum eavesdropper, \emph{even if the input randomness
was generated by that same eavesdropper}!
  The notion of Input Security may be of independent interest to other areas
such as device-independent quantum key distribution.
