For me not fully reproducible with LO 4.4.3.2, Win 8.1 as described in the bug report.  
If I insert the short sample text then it is not expanded.  But if I insert a longer text (over several lines) then it is expanded.  But also in the longer text case the last line is not expanded.

oh but you have reproduced it.

If the first line is the last line then that too should be expanded.

1. New Text Document.
2. Type “this is a test”.
3. Click on Justified on the toolbar.
Result:
Text doesn't expand which is correct.
4. Right click on paragraph → Alignment tab, set Last line to “Justified” (notice the preview) and OK.
Result:
Text expands to fill the width of the page.

If you continue to add text to that line until it goes to the next line then that becomes the last line and if it is short then it will expand to fill the width of the page.  Change the last line back to "Left" and compare the behaviour to the text box.

You can do interesting things like set last line to justified and add large equidistant-from-the-centre indents before and after text so you can have a one line title/slogan centred and justified on the page.

Both "Justified" and "Centred" on the last line do not work in text boxes either on a one line paragraph or on the Nth line.

(In reply to Gordo from comment #2)
> oh but you have reproduced it.
> 
> If the first line is the last line then that too should be expanded.
> 
> 1. New Text Document.
> 2. Type “this is a test”.
> 3. Click on Justified on the toolbar.
> Result:
> Text doesn't expand which is correct.
> 4. Right click on paragraph → Alignment tab, set Last line to “Justified”
> (notice the preview) and OK.
> Result:
> Text expands to fill the width of the page.
> 
> If you continue to add text to that line until it goes to the next line then
> that becomes the last line and if it is short then it will expand to fill
> the width of the page.  Change the last line back to "Left" and compare the
> behaviour to the text box.
> 
> You can do interesting things like set last line to justified and add large
> equidistant-from-the-centre indents before and after text so you can have a
> one line title/slogan centred and justified on the page.
> 
> Both "Justified" and "Centred" on the last line do not work in text boxes
> either on a one line paragraph or on the Nth line.

Yes, the last line issue I could reproduce.
But I could not reproduce "To ensure that this bug was not restricted to a single line paragraph, I typed more text into the Text Box and the result was the same".  Because if I had a longer text then all the text was justified, besides the last line.  Or is it a misunderstanding from my side and you meant with your bug report only this last line issue?

This:

"To ensure that this bug was not restricted to a single line paragraph, I typed more text into the Text Box and the result was the same."

meant that maybe it was only happening for the single line paragraph, which was the last line, so I added text to create a second line, which was now the last line and the second line did not expand.  The first line expanded correctly.

I thought maybe it was not working just for a single line paragraph and that if there was a second line then the second line would expand but it did not.

*** Bug 146464 has been marked as a duplicate of this bug. ***

Reproduced in:

Version: 7.3.0.0.beta1 / LibreOffice Community
Build ID: 436f14c25ec1847646b953cf13d0db4f7ca3be57
CPU threads: 8; OS: Linux 5.4; UI render: default; VCL: gtk3
Locale: en-AU (en_AU.UTF-8); UI: en-US
Calc: threaded

and:

Version: 7.4.0.0.alpha0+ / LibreOffice Community
Build ID: 7fe2ce55ab86cc7a32850fdf504e368c535949c3
CPU threads: 8; OS: Linux 5.4; UI render: default; VCL: gtk3
Locale: en-AU (en_AU.UTF-8); UI: en-US
Calc: threaded

*** Bug 131058 has been marked as a duplicate of this bug. ***

I pushed a request to gerrit to hide related options until they are fixed, please review it :-)
https://gerrit.libreoffice.org/c/core/+/129807