How do I find out what version of WordPress is running?

Unless he edited some code to delete this, you should be able to view source on the site and look for this meta tag:

<meta name="generator" content="WordPress 2.7.1" /> 

That will give you the version.

On the Admin Panel Dashboard, you can find a box called "Right Now". There you can see the version of the WordPress installation. I have seen this result in WordPress 3.2.1. You can also see this in version 3.7.1

UPDATE:

In WP Version 3.8.3

In WP Version 3.9.1 Admin Side, You can see the version by clicking the WP logo which is located at the left-top position.

You can use yoursitename/readme.html

In the WordPress Admin Footer at the Right side, you will see the version info(Version 3.9.1).

You can get the WordPress version using the following code:

<?php bloginfo('version'); ?>

The below file is having all version details

wp-includes/version.php   

Update for WP 4.1.5

In WP 4.1.5, If it was the latest WP version in the footer right part, it will show the version as it is. If not, it will show the latest WP version with the link to update.

Check the below screenshot.

Look in wp-includes/version.php

/**
 * The WordPress version string
 *
 * @global string $wp_version
 */
$wp_version = '2.8.4';