It is possible to way b .htaccess file could be written to ensure that

http://www.mysite.com/about 

loads about.php

and you have to any file?

I attempted

RewriteEngine On
RewriteRule ^([0-9A-Za-z]+)$ $1.php

however it does not appear to operate.

This is actually the most typical rule (making use of mod_rewrite -- make certain it's loaded and enabled) -- it'll make sure that such .php file does exist before spinning (yes, it's a little reduced but safer):

Options +FollowSymLinks
RewriteEngine On

# add .php file extension
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_FILENAME}\.php -f
RewriteRule ^(.+)$ $1.php [L]

Alternatively only use this:

Options +MultiViews

This can switch on "content settlement". However it has some cons.

For instance: let us assume you've hello.html &lifier hello.php inside your website root folder. Should you request example.com/hello, with this option enabled Apache will search for alternative names (same title but different extensions) and can serve either hello.html or hello.php (I am unable to tell which is going to be preferred).

But when you simply have 1 file with your unique title (e.g. hello.php ONLY) then no problems here whatsoever.