In addition to checking for the CSRF token as a POST parameter, the Laravel VerifyCsrfToken middleware will also check for the X-CSRF-TOKEN request header. You could, for example, store the token in a "meta" tag:
<meta name="csrf-token" content="{{ csrf_token() }}">
Once you have created the meta tag, you can instruct a library like jQuery to add the token to all request headers. This provides simple, convenient CSRF protection for your AJAX based applications:
$.ajaxSetup({ headers: { 'X-CSRF-TOKEN': $('meta[name="csrf-token"]').attr('content') } });