I found the following code to stop the sanitisation of HTML elements from the User > Edit User > Biographical Information text area.
// Disable WordPress sanitization to allow more than just $allowedtags from /wp-includes/kses.php.
remove_filter( 'pre_user_description', 'wp_filter_kses' );
// Add sanitization for WordPress posts.
add_filter( 'pre_user_description', 'wp_filter_post_kses' );
…which does allow me to add <p> or <br> elements to the text box, and doesn’t strip them when saved but it won’t show on the front-end in the post. Still a mass of text.