Set a sitewide max shipping amount

max_shipping.php

<?php
/**
 * Function to set a max cap on the shipping rates.
 */
function my_max_limit_shipping_rates_function( $rates, $package ) {
	$methods = WC()->shipping()->get_shipping_methods();
	$shipping_limit = 15; // The maximum amount would be $15
	// Loop through all rates
	foreach ( $rates as $rate_id => $rate ) {
		// Check if the rate is higher then a certain amount
		if ( $rate->cost > $shipping_limit ) {
			$rate->cost = $shipping_limit;
			// Recalculate shipping taxes
			if ( $method = $methods[ $rate->get_method_id() ] ) {
				$taxes = WC_Tax::calc_shipping_tax( $rate->cost, WC_Tax::get_shipping_tax_rates() );
				$rate->set_taxes( $method->is_taxable() ? $taxes : array() );
			}
		}
	}
	return $rates;
}
add_filter( 'woocommerce_package_rates', 'my_max_limit_shipping_rates_function', 10, 2 );