WP-Admin not working properly at WordPress multisite with subdirectories

If you installed WordPress Multisite starting with version 3.0 to 3.4.2, you’ve got the correct .htaccess file contents.

However, if you started with a newer version (3.5 or higher)—and I’m assuming you did, if you’ve just installed WordPress Multisite recently—your .htaccess file should look like this:

RewriteEngine On
RewriteBase /
RewriteRule ^index\.php$ - [L]

# add a trailing slash to /wp-admin
RewriteRule ^([_0-9a-zA-Z-]+/)?wp-admin$ $1wp-admin/ [R=301,L]

RewriteCond %{REQUEST_FILENAME} -f [OR]
RewriteCond %{REQUEST_FILENAME} -d
RewriteRule ^ - [L]
RewriteRule ^([_0-9a-zA-Z-]+/)?(wp-(content|admin|includes).*) $2 [L]
RewriteRule ^([_0-9a-zA-Z-]+/)?(.*\.php)$ $2 [L]
RewriteRule . index.php [L]

I’m assuming you’re using Multisite in Subfolder mode, since in your question your URL is of the form example.com/site/wp-admin. (Edit: Also I just noticed that it’s definitely Subfolder per your posted wp-config.php file.)

See the Codex page on .htaccess for details.