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.)
def apply — even if the committee has begun reviewing, it's customary for the chair of the search to peruse any new applicants that come in and forward only the most promising apps on to the rest of the committee for last minute review. another possible option is that the initial candidates invited out don't make muster and the committee considers going back to the pool… they'll be most likely to look at the applications that have been newly submitted.