  MoEDAL is designed to identify new physics in the form of long-lived
highly-ionising particles produced in high-energy LHC collisions. Its arrays of
plastic nuclear-track detectors and aluminium trapping volumes provide two
independent passive detection techniques. We present here the results of a
first search for magnetic monopole production in 13 TeV proton-proton
collisions using the trapping technique, extending a previous publication with
8 TeV data during LHC run-1. A total of 222 kg of MoEDAL trapping detector
samples was exposed in the forward region and analysed by searching for induced
persistent currents after passage through a superconducting magnetometer.
Magnetic charges exceeding half the Dirac charge are excluded in all samples
and limits are placed for the first time on the production of magnetic
monopoles in 13 TeV $pp$ collisions. The search probes mass ranges previously
inaccessible to collider experiments for up to five times the Dirac charge.
