  The 0-instanton solution of Painlev\'e I is a sequence $(u_{n,0})$ of complex
numbers which appears universally in many enumerative problems in algebraic
geometry, graph theory, matrix models and 2-dimensional quantum gravity. The
asymptotics of the 0-instanton $(u_{n,0})$ for large $n$ were obtained by the
third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the
$k$-instanton solution of Painlev\'e I is a doubly-indexed sequence $(u_{n,k})$
of complex numbers that satisfies an explicit quadratic non-linear recursion
relation. The goal of the paper is three-fold: (a) to compute the asymptotics
of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the
Riemann-Hilbert method, (b) to present formulas for the asymptotics of
$(u_{n,k})$ for fixed $k$ and to all orders in $1/n$ using resurgent analysis,
and (c) to confirm numerically the predictions of resurgent analysis. We point
out that the instanton solutions display a new type of Stokes behavior, induced
from the tritronqu\'ee Painlev\'e transcendents, and which we call the induced
Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new
phenomena not seen in 0 and 1-instantons, and their enumerative context is at
present unknown.
