Create a custom xml template for the button ie

<templates>

    <t t-name="MyModule.TopButtonMenu">
        <li class="o_mail_navbar_item">
            <button>My Button</button>                            
        </li>
    </t>
</templates>

And place it at the top menu with javascript

odoo.define('MyModule.TopButton', function(require) {
    "use strict";
​var SystrayMenu = require('web.SystrayMenu');

var TopButton = Widget.extend({
    template:'MyModule.TopButtonMenu',
    events: {
        "click": "on_click"
    },

on_click: function (event) {
//do action
}
});

SystrayMenu.Items.push(TopButton);

});

​