Suppose I dug an open pit mine, 1 km diameter and straight down, how far down could I before I literally could not go any deeper and what would happen at that limit?

 Palabora pit in South Africa before and after abandonment and partial collapse.

The steepest divider point an open pit can uphold without imploding is around 60°, so with a range of 500m (measurement of 1km) your pit will reach as far down as possible at around 870m. On the off chance that you need to go further, you need to extend the pit. On the off chance that you need to go more extreme, it will probably fall.