  We compute the n-point two-loop form factors of the half-BPS operators
Tr(phi_{AB}^n) in N=4 super Yang-Mills for arbitrary n >2 using generalised
unitarity and symbols. These form factors are minimal in the sense that the
n^{th} power of the scalar field in the operator requires the presence of at
least n on-shell legs. Infrared divergences are shown to exponentiate as for
amplitudes, reproducing the known cusp and collinear anomalous dimensions at
two loops. We define appropriate infrared-finite remainder functions and
compute them analytically for all n. The results obtained by using the known
expressions of the integral functions involve complicated combinations of
Goncharov multiple polylogarithms, but we show that much simpler expressions
can in fact be derived using the symbol of transcendental functions. For n=3 we
find a very compact remainder function expressed in terms of classical
polylogarithms only. For arbitrary n>3 we are able to write all the remainder
functions in terms of a single compact building block, expressed as a sum of
classical polylogarithms augmented by two multiple polylogarithms. The
decomposition of the symbol into specific components is crucial in order to
single out a natural combination of multiple polylogarithms. Finally, we
analyse in detail the behaviour of these minimal form factors in collinear and
soft limits, which deviates from the usual behaviour of amplitudes and
non-minimal form factors.
