  If the symmetry breaking responsible for axion dark matter production occurs
during the radiation-dominated epoch in the early Universe, then this produces
large amplitude perturbations that collapse into dense objects known as axion
miniclusters. The characteristic minicluster mass, $M_0$, is set by the mass
inside the horizon when axion oscillations begin. For the QCD axion $M_0\sim
10^{-10}M_\odot$, however for an axion-like particle $M_0$ can approach
$M_\odot$ or higher. Using the Press-Schechter formalism we compute the mass
function of halos formed by hierarchical structure formation from these seeds.
We compute the concentrations and collapse times of these halos and show that
they can grow to be as massive as $10^6M_0$. Within the halos, miniclusters
likely remain tightly bound, and we compute their gravitational microlensing
signal taking the fraction of axion dark matter collapsed into miniclusters,
$f_{\rm MC}$, as a free parameter. A large value of $f_{\rm MC}$ severely
weakens constraints on axion scenarios from direct detection experiments. We
take into account the non-Gaussian distribution of sizes of miniclusters and
determine how this effects the number of microlensing events. We develop the
tools to consider microlensing by an extended mass function of non-point-like
objects, and use microlensing data to place the first observational constraints
on $f_{\rm MC}$. This opens a new window for the potential discovery of the
axion.
