I'd say the exception is when `n` is under about 10, and is counting some sort of hardware constrained thing (e.g. some operation over all CAN interfaces pesent on an OBDII connector can be O(n^(2)) since n will always be between 1 and 4). If you wouldn't have to physically replace hardware for `n` to increase, you really need to avoid n^2 operations. And even then consider them carefully, perhaps explicitly failing if `n` gets too big to allow for noticing rework is needed before new hardware hits the field.
Real production use cases absolutely have small n. You don't hear about them, because it's very hard for them to cause issues. Unless the use case changes and now the n is not small anymore and nobody noticed the trap.
I have an app that's been running an O n^2 algorithm in "production" (free open source app used by various communities) for about half a year now.
It's been fine because "n" is "number of aircraft flying in this flight simulator" - and the simulator's engine starts to fail above around 2000 anyway. So even in the worst case it's still going to run within milliseconds.
