  The cross-section of associated $J/\psi$-$\psi(2S)$ production in
proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=13$ TeV is
measured using a data sample corresponding to an integrated luminosity of 4.2
fb$^{-1}$, collected by the LHCb experiment. The measurement is performed for
both $J/\psi$ and $\psi(2S)$ mesons having transverse momentum
$p_{\text{T}}<14$ GeV/$c$ and rapidity $2.0<y<4.5$, assuming negligible
polarisation of the $J/\psi$ and $\psi(2S)$ mesons. The production
cross-section is measured to be $4.5\pm0.7\pm0.3$ nb, where the first
uncertainty is statistical and the second systematic. The differential
cross-sections are measured as functions of several kinematic variables of the
$J/\psi$-$\psi(2S)$ candidates. The results are combined with a measurement of
$J/\psi$-$J/\psi$ production, giving a cross-section ratio between
$J/\psi$-$\psi(2S)$ and $J/\psi$-$J/\psi$ production of
$0.274\pm0.044\pm0.008$, where the first uncertainty is statistical and the
second systematic.
