Following the extremely successful operation of the Large Hadron Collider at CERN during its first years of operation, the ATLAS and CMS collaborations announced the discovery of a new particle compatible with the Higgs boson in July 2012, finishing the complement of particles in the Standard Model. However, no other new particle was even hinted at, confounding predictions of an early discovery of Supersymmetry that would resolve the hierarchy problem and retain low fine-tuning. This thesis presents two searches for new phenomena that would have evaded early efforts. The first, using the full 8 TeV dataset collected in 2012 by ATLAS, targets heavy particles decaying to charm quarks and non-interacting particles through the use of a recently developed charm-jet identification algorithm. No excess is observed over the Standard Model predictions, so results are interpreted in the context of supersymmetric charm production. This is constrained to the highest masses so far, with scharms excluded at the 95% confidence level for masses up to 550 GeV with neutralino masses below 200 GeV. The second search makes use of the 13 TeV dataset collected in 2015, and looks for heavier particles which are produced with far higher cross section at this increased collision energy. It is sensitive to particles decaying via cascades to large numbers (= 7 to = 10) of hadronic jets, with sensitivity enhanced through requirements on the number of jets identified as arising from bottom quarks. Again, no excess is observed, and so limits are set on gluino production in two cascade decay scenarios, with a significant increase in reach obtained over 2012 searches for the same final state - gluino masses below 1400 GeV are excluded in the models considered.