Xenon 136, apparently, has a half-life of 2.11×1021 years. This strikes me as a humongously long time to run an experiment, clocking in at about 11 orders of magnitude longer than the age of the universe.
This question puts some numbers in and shows that it's really quite reasonable to determine half-lives in the range of 109 years by simply counting the number of decays in a decently short amount of time. However, if you do equivalent numbers for xenon 136 you get numbers which push this sort of thing much harder: 1g of 136Xe will produce about 1.5 decays in a year, and this already occupies 160 ml in standard gas phase conditions. To get reasonable statistics, you will need either a lot of isotopically pure xenon or a long time of very efficient, background-free detection, or (probably) both.
So, 1021 years is still sort of reasonable, but push in a few more zeros and you start having an untenable experiments in your hands. And, indeed, a quick browse through Wolfram Research's curated IsotopeData suggests that the longest known half-life is that of 130Te, at around 5×1023 years.
I seem to have run out of questions and answered the ones I initially had, so instead I'll push this a bit further: are these walls hard? That is, can we plausibly measure longer half-lives? Are there current experiments trying to do so? Is 130Te indeed the longest we know? Given the difficulty in establishing that it decays at all, the different 'observationally stable' isotopes of tellurium seem more observational and less stable than they do at first sight.