Bulbapedia says the chance increases to 1/1024 at a chain greater than or equal to 70, but this dataminer claims the chances increase much more than that at multiple intervals. From my experience chaining, I am much more inclined to believe the dataminer.
The dataminer's grid shows that the shiny chance increase is caused by the game rerolling the PID at later parts of the chain if it isn't shiny; 5 times at a chain of 11-20, 9 times at a chain of 21-30, and capping out at 13 rerolls at a chain length greater than 30. Additionally, if the player has the Shiny Charm, the game rerolls two more times, regardless of chain length. In short, this means that, after a chain length of 31, the shiny odds cap out at a 1/315 chance (1/273 with shiny charm).
2021 edit: For anyone who is still unsure whether the shiny odds are 1/1024 after 70 or 1/315 after 31, I can now very confidently say that the 1/315 after 31 statistic is accurate. A few months ago, I recorded the chain lengths for 100 shiny Exeggcutes with charm. The total encounters for all of them ended up being 30,114 encounters, giving an average length of 301 per shiny. This is slightly higher than the odds with charm (1/273), which makes sense given that the odds are worse at the beginning.