I need a 'fast-entry' way of entering a log of potentially billable events as they occur

use onenote for this

It isn't accessible enough. I need it to live in an open state down in the bottom right corner of my screen or other suitable area and be completely dedicated to entering some text and pressing enter.

click on the icon on the taskbar and add your note

That's not the objective. I need it to be open and always on top and dedicated to this purpose.

You could follow the advice (found here) to use AutoHotkey and set-up a hotkey to keep any window you want to stay on top. e.g. make Notepad stay on top of all windows.

This way you can use whatever text editor you already like to use, but with the added stay-on-top capability.

I'm looking for a way to enter a newly completed task, then enter, so the text entry field would always need to be empty and available.

Note that One Note has a "always on top " feature built-in. .   It's on the view menu (make sure the window is stretched out as it's the last entry on the ribbon and you won't see it if it's not wide enough).

Also, look at using a table in one note.   As soon as you hit return, it will advance to the next row of the table.   That should be pretty close to what you want.

Jim.

"enter a newly completed task, then enter" 
and then that entry goes where?   
                     to your "billing report "?
                     What is your "billing report"? e.g. Quickbooks? Excel? Word Doc? Access table?
then
          the text entry field is cleared to be ready for another entry
and
         this "stays on top"

something to your exact specifications may need to be built

I'll have to look at the OneNote solution and see if it will work but I'm just not envisioning it.

I just want to be able to put down what tasks been completed during the day as they get finished with his little complexity as possible. I don't even need a timestamp

Open up One Note.

Insert a 1 row, 1 column table.

Type something in the first row, tab/return out.  A new rowwill automatically be added and you'll be on the second row ready to type.   I will continue to add new rows as needed.

You can add an additional column if you want to keep track of time as well as a comment.   Also note that you can easily convert that table to an Excel spreadsheet if you need to work with the data.   You also can apply formatting to tables.

Last, on the ribbon, on the view tab, the "Always on top" is on the far right.

You also can organize in One Note by job, day, or whatever (ie. start a new table each day). 

With something off the shelf, I don't see how you can come much closer to what you were asking for, unless we are missing the point on what you are trying to do.

Jim.

I have tried to further illustrate the description above and in this screencast where I tried the OneNote option https://www.screencast.com/t/7lZY6lyW7