I also don't have the hamburger icon on mobile, so it's not an isolated issue
Viewing post in Where is the search bar on this site?!
That hamburger is only on / and only if you have a narrow display, like a mobile phone. You can simulate this by changing width of the browser window.
Seems odd, that it only shows there. If you are already on /games there is no search box for a narrow screen. There is also none if you already have searched.