There's a job I'd like to apply for but the "for full consideration apply by" date has just passed. The school still seems to be taking applications, though. Should I send one in anyway? (I could get the app in fairly quickly but I'd have to give my references at least a week if not two.) Assuming the answer is apply, are not so late applications discounted in anyway?
I'd say apply. There might be some places/departments/committees that won't consider late applications, but I know of some that have/do. And it's not uncommon for rec letters to get in a little late. It's one of those cases where worst case scenario is you don't get an interview. So what do you have to lose?
I was offered a job once when my application was a few days late. If the deadline was 10/15 I wouldn't worry about it and definitely apply. If it was 10/1, you can try, but your chances are probably not so great if it's a remotely desirable position and not too specific in terms of research area. (On the flip side, in that same application cycle I sent in an application 30 minutes late for a 10/1 [early!] deadline when it was due over a weekend and it was due to an e-mail address rather than a generalized system. I figured they wouldn't really cut off applications until Monday. I was wrong. I got a notification about 3 weeks later that my application was not considered because they stuck to the deadline and they had received over 500 applications. The position was at a mid-tier liberal arts school / regional university on the east coast in a non-city but non-remote location with a 3-3 load. Not the most amazing position, but it was open area, so it was apparently very popular.)