We report the discovery of WTS-2 b, an unusually close-in 1.02-d hot Jupiter (MP = 1.12MJ, RP = 1.30RJ) orbiting a K2V star, which has a possible gravitationally bound M-dwarf companion at 0.6 arcsec separation contributing ˜20 per cent of the total flux in the observed J-band light curve. The planet is only 1.5 times the separation from its host star at which it would be destroyed by Roche lobe overflow, and has a predicted remaining lifetime of just ˜40 Myr, assuming a tidal dissipation quality factor of Q_{*}^'}=106. Q_{*}^'} is a key factor in determining how frictional processes within a host star affect the orbital evolution of its companion giant planets, but it is currently poorly constrained by observations. We calculate that the orbital decay of WTS-2 b would correspond to a shift in its transit arrival time of Tshift ˜ 17 s after 15 yr assuming Q_{*}^'}=106. A shift less than this would place a direct observational constraint on the lower limit of Q_{*}^'} in this system. We also report a correction to the previously published expected Tshift for WASP-18 b, finding that Tshift = 356 s after 10 yr for Q_{*}^'}=106, which is much larger than the estimated 28 s quoted in WASP-18 b discovery paper. We attempted to constrain Q_{*}^'} via a study of the entire population of known transiting hot Jupiters, but our results were inconclusive, requiring a more detailed treatment of transit survey sensitivities at long periods. We conclude that the most informative and straightforward constraints on Q_{*}^'} will be obtained by direct observational measurements of the shift in transit arrival times in individual hot Jupiter systems. We show that this is achievable across the mass spectrum of exoplanet host stars within a decade, and will directly probe the effects of stellar interior structure on tidal dissipation.