Unfortunately not allowed per ICANN's contract for new gTLDs. Though even if it were, it probably wouldn't be a good idea to actually use it because support for first level globally resolving strings is quite limited across a variety of applications and OS stacks. Oftentimes you have to append the trailing canonical dot to get it to even try to make the correct globally resolving DNS request, and that's just not good/consistent UX.
It's allowed with the Handshake protocol.
Why should it though? And what content would be there? Note that there is an https://about.google
about.google is about google. blog.google is a blog maintained by google.
So google.google should google 'Google' on google for you.
... but that's not even a good joke, and wouldn't be worth the operational costs of launching it. com.google was a joke worthy of launching: https://www.businessinsider.com/search-google-backwards-secr... (though not worthy of maintaining once April Fools was over)
Since I am not familiar with the costs involved, can you say how much would it cost to just redirect google.google to https://www.google.com/search?q=google ?
Independent of the costs (which are higher than you might expect), it's just not good practice to create a whole slew of random redirects "just because". There's not a lot of random inbound traffic to "google.google" that plausibly needs redirecting. Indeed, you'd be causing the opposite problem, which is that by creating the redirect, you're now encouraging people to use it, and now you have an additional thing to support that you never really wanted in the first place. Remember that *.google isn't "just" a redirect or a subdomain -- it's a bona fide domain name, requiring registration, ongoing management thereof, an SSL certificate, configuration of automatic renewal thereof, DNS configuration, uptime monitoring, etc. And that's before you even get into any potential concerns re: the public suffix list, HSTS configuration/preloading, and other more esoteric concerns. It ends up not being worth it unless you have a real site that you actually want to launch, or at the minimum, a redirect that you really want to exist that you plan on using widely.