Application Insights Delay?

Generally raw examples of your data should be available within couple of minutes from the time you send it, and aggregated data takes about 5-10 minutes to appear. Also when we are experiencing a processing delay we display a banner on the Overview page in Application Insights in the portal as on the screenshot below.

If you saw 40 minutes delay seeing your data this was either the case of ongoing issue with the processing pipeline, in which case a message should have been shown (and if not, it is a detection problem on our side), or, as we are often seeing, there could have been a configuration problem with your application that was later addressed.

Agree with the comments in the accepted answer that real-time logging is a absolute requirement of an enterprise system. Even the Portal says the following on the Monitor section of the Azure Functions blade:

This appears to be due to metric aggregation. However I've just been shown Application Insights' Live Metrics Stream by a colleague. It has 1-second latency, which is probably what most readers of this question are after and thought worth sharing.