Is there a very very basic example of simply outputting PHP with a plugin to Twig? For arguments sake let's say I wanted a plugin that is a basic output: 

&lt;?php echo '&lt;p&gt;Hello World&lt;/p&gt;'; ?&gt;


So all I am wanting to output is that, and I want that output to be available via twig (ie: {{ helloworld }}

Not passing any information, not doing anything with the craft database, etc. Just very basic.
Have a look at plugin template variables. You can simply create a method in your plugins variable file that returns anything you want to the template, which would be accessible via {{ craft.myplugin.customVariable }}.

&lt;?php
namespace Craft;

class MypluginVariable
{
    public function customVariable($someProperty = 'defaultValueIfNotDefined')
    {
        return 'Whatever you want!';
    };
}