The Large Hadron Collider has many years of operations ahead of it, but it is increasingly looking likely that those experiments will continue to confirm that they have found a Standard Model Higgs boson, with this hypothesis becoming more and more certain with each coming year, and that they have not found any deviations from the Standard Model predictions, with the relevant parameters being pinned down with somewhat greater precision over time. It is also quite likely that exclusion areas from the SUSY parameter space and of other beyond the Standard Model theories will be increasingly ruled out.
There will still be open questions in neutrino physics, since the LHC is ill suited to reveal the neutrino's mysteries (we lack basic parameters like the neutrino masses in absolute terms and accurate measurements of the values of the PMNS matrix), and there will still always be the possibility that some new physics is lurking at an energy scale beyond that observable at the LHC after a full run.
As I've noted before, a particularly important parallel track of fundamental physics research involves the ongoing efforts to see, or exclude down to a certain production rate, neutrinoless double beta decay - I doubt that it is out there or that neutrinos have Majorana mass (which an obserbation of neutrinoless double beta decay would imply). Equally important, a failure to find neutrinoless double beta decay below a sufficiently low rate of production, combined with a lack of direct detection of supersymmetric particles in the LHC's planned run, pretty much closes the door on all versions of SUSY that make sense and a fortori, falsifies String Theory.
If all of the leading alternatives to the Standard Model with a single Standard Model Higgs boson and Dirac mass neutrions are ruled out and the remaining neutrino related constants of the Standard Model are established more precisely, which could happen before my kids are old enough to take the early graduate school level classes where physis like this is taught, the possibility of a new physics "desert" all of the way up to the Grand Unification scale (which will probably never be tested in man made experiments) starts to look very plausible.
The one point of experimental evidence which has overconstrained theory, ruling out every possible explanation in one way or another, is the phenomena attributed to dark matter. In the range of particle properties that could be consistent with what astronomers observe, high energy physicists have pretty much ruled out ever possible candidate particle (with the possible exception of sterile neutrinos). But, modifications to the laws of gravity in weak fields, which can predict many of the same phenomena, also seem to be contradicted or rendered incomplete by seem key astronomy observations like the Bullet Cluster collision.
There is also progress being made in ironing out some of the quirks in the Standard Model. Folks like Italian physicist Macro Frasca and Russian physicists I.M. Suslov are making inroads into fixing some of the technical anomolies key sets of equations like the Landau pole of Quantum Electrodynamics and non-trivial fixed points in pure Yang-Mills theory (which is essentially strong force physics before quarks are introduced into the mix).
For work day to day at high energy physics experiments figuring out Standard Model backgrounds this doesn't matter too much, since the quirks manifest in the extreme high and low energy regimes of strong force physics that can't be directly observed due to lack of experiment scale and confinement respectively, and in QED has extremely short distances and extremely high energies. But, these developments are critical in theoretical tasks like unifying the Standard Model forces into some kind of grand unified theory or even, if one includes quantum gravity as well, as so called "Theory of Everything." They also give us greater confidence that existing equations are accurate up to arbitrarily high energy scales and allow us to revisit conjectures made with incorrect assumptions about the Higgs boson and the beta functions that govern the running of the respective coupling constants.