Please don't question my curiosity.
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 shinychance 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.