You can only reasonably do what you are asking with MultiViews.

For example, as simple as:

Options +MultiViews

You need to remove your existing mod_rewrite directives.

Now, a request for example.com/horses will be correctly routed to /horses.txt, or whatever file extensions you are using. MultiViews uses mod_negotiation.

This isn't so easy to do with mod_rewrite, since you need to test each file extension in turn in order to work out what file you need to rewrite back to in order to route the request correctly. eg. Should a request for example.com/horses route to /horses.txt or horses.jpg? MultiViews does this comparison for you.

I would like a to do a rewrite rule that removes all extensions

Although, you need to actually remove the file extension in the HTML source. This isn't something you do in .htaccess, unless you need to preserve SEO or backlinks that have already linked back to the old URLs.

UPDATE: Perhaps I wasn't clear enough, I would like the url to display without the extension even if it is linked to it, or to go to that file if linked without the extension

Well, you need to actually remove the file extension on all your internal links. You can issue a "redirect" in .htaccess to remove the extension for the benefit of search engines and 3rd party links - but if you rely on this for your internal links then it will potentially slow users and your site as you are doubling the number of requests hitting your server.

To remove the file extension for direct requests (SEO / 3rd party links), you could do something like this:

RewriteEngine On

RewriteCond %{ENV:REDIRECT_STATUS} ^$
RewriteCond %{REQUEST_FILENAME} -f
RewriteRule ^([^.]+)\.[\w]{2,4}$ /$1 [R=302,L]

This does assume that the only dot in the URL-path is the one that delimits the file extension.

The difficult part is then internally rewriting the request back to the underlying file with an extension - that's where MultiViews comes in (first part of my answer).