class-remove-h1-format.php

<?php
/* File level DocBlock here - includes/class-remove-h1-format.php */

/* Class level DocBlock here */
class Remove_H1_Format {
	/*
	 * Modify TinyMCE editor to remove H1.
	 */
	public function remove( $init ) {
		// GJ: No need for the filter - if they don't want it running, deactivate the plugin.

		// Add block format elements you want to show in dropdown
		// GJ: I'd probably go for str_replace() here, so that you don't trump over
		// other filters modifying the same block_formats.
		$init['block_formats'] = 'Paragraph=p;Heading 2=h2;Heading 3=h3;Heading 4=h4;Heading 5=h5;Heading 6=h6;Address=address;Pre=pre';

		return $init;
	}

}

remove-h1-format.php

<?php
/**
 * Plugin Name:  Remove H1 Format
 * Plugin URI:   http://calliaweb.co.uk/modify-tinymce-editor/
 * Description:  A simple plugin to remove the H1 format from the WordPress TinyMCE editor.
 * Version:      1.0.0
 * Author:       Jo Waltham
 * Author URI:   http://calliaweb.co.uk/
 * License:      GPL-2.0+
 * License URI:  http://www.gnu.org/licenses/gpl-2.0.html
 */

// Exit if accessed directly
defined( 'WPINC' ) or die;

include plugin_dir_path( __FILE__ ) . 'includes/class-remove-h1-format.php';

$remove_h1_format = new Remove_H1_Format;
add_filter( 'tiny_mce_before_init', array( $remove_h1_format, 'remove' ) );