I have used ocenaudio, which I like that it has the current timestamp while playing. Audacity has a timebar, which you can get the approximate timestamp for, however getting a more accurate timestamp is useful for sharing audio on sites such as Soundcloud and xeno-canto (bird sounds) and for listeners to click on the timestamp to go straight to it.
By “current timestamp”, do you mean “clock time” (the time on my mantelpiece clock), or do you mean “current play position, relative to the start of the track”?
Audacity has a time-line … Timeline - Audacity Manual
and an “Audio Position” counter …
Audacity doesn’t have a Clock Time recorder. It’s one of the reasons it makes a terrible surveillance recorder.
There is a completely wacko way to do this. Get a timecode generator and jam it into one side of the stereo tracks. The show is on the other side (in mono).
With a time code reader, you can reliably find 1430 this afternoon just by playing that track.
It’s been a while. I think that’s one minute before midnight (watch your ears).
Audacity also has an option to include the “clock time” in the name of a newly added track, so that you have a record of when the recording started.