What are the most important questions we need experiments to answer in physics right now?
Here is my list:
1. Discover or reduce the experimental bound on the maximum rate of neutrinoless double beta decay (I suspect it will not be found and this rules out many BSM theories).
2. Continue to place experimental bounds on proton decay rates (I suspect that it does not happen and this rules out many BSM theories).
3. Determine the CP violating phase of the PMNS matrix that governs neutrino oscillations (anybody's guess, but probably not zero).
4. Determine the absolute masses of the three neutrino mass eigenstates and whether the neutrinos have a "normal", "inverted" or "degenerate" hierarchy of masses (probably "normal" and very small).
5. Refine the precision with which we know the mass of the top quark (relevant to making relationships between Standard Model experimentally measured masses convincing).
6. Refine the precision with which we know the properties of the Higgs boson, particularly its mass (relative to making relationships between experimentally measured masses convincing) and any possible other variation from the Standard Model prediction (something that will probably not be found).
7. Complete the second phase of the analysis of the Planck satellite data (relevant to distinguishing between quintessence and the cosmological constant, with the latter more likely supported, and to ruling out possible cosmological inflation theories with the simplest theories currently favored).
8. Continue the search for glueballs, tetraquarks and pentaquarks - all of which are theoretically possible in QCD but not yet definitively observed (everything that is not forbidden in physics is mandatory, so the absence or presence of these phenomena have great importance by adding new QCD rules).
9. Tighten experimental boundaries on the masses of the five lightest quarks (this would allow for the proof or disproof of extensions of Koide's rule for quarks - the precision of these measurements is currently very low).
10. Conduct more astronomy observations that constrain the possible parameters of dark matter or any alternative theory that explains phenomena attributed to dark matter (dark matter is the single most glaring missing piece of modern physics), including measurements of ordinary "dim matter".
11. Experiments to reconcile discrepancies between muonic hydrogen and ordinary hydrogen's properties (probably due to imprecision in ordinary hydrogen measurements).
12. Improve the precision of QCD calculations that form backgrounds for other experimental measurements giving all other measurements at the LHC and elsewhere more statistical power.
LHC, the current physics show horse, is pertinent only to numbers 5, 6, 8 and 9. Progress on 5, 6 and 9 is likely to be very incremental after the next couple of years.
Experimental searches that I deemed less worthy, because I think they are less likely to be fruitful include:
1. Searches for supersymmetric particles or additional Higgs bosons (SUSY is increasingly ill motivated).
2. Searches for additional compact dimensions.
3. Searches for W' and Z' particles.
4. Searches for fourth generation Standard Model particles (basically ruled out already).
5. Direct dark matter detection experiments (the cross-section of interaction is too small to be likely to find anything with conceivable near term experiments as other data favors something akin to sterile neutrino 2 keV warm dark matter).
Alternative suggestions in the comments (with justifications) are welcome.