For my phone, I'm already logged in and never get any future challenges. I needed the Yubikey when I first logged into my phone, but after that the phone has been authenticated. If I unlink my phone to my Google account I'll need the Yubikey again, but I don't normally do that. So normally I don't carry a Yubikey with me, like when I go to the store and what not.

That said, I do keep a Yubikey with me in my bag when I travel in case my phone breaks and I need to authenticate into a new device. I do take a Yubikey with me going to and from the office as there are other services and platforms which do challenge my Yubikey more often.

I have a yubikey on my keyring. It's superior to sms 2fa in everyway. Its almost impossible to damage a yubikey- phones can easily be broken or stolen. You can have multiple keys linked to your account- Google only let's you have one phone. Yubikey can't be sim swapped. Never needs to be charged or have cell reception, no problems with sites not accepting international phone numbers.

The only downside is that Google is the only site I used that supports it.

I keep one on my keychain in my pocket and one at home in a fireproof box, plus a backup one that I haven't even opened next to the backup so if I lose the keychain one I have another ready to go as my "new backup".

As others have commented, on your phone you rarely ever need to authenticate, so I keep mine at home.

If you buy a Titan Key you get two (USB-A, USB-C), so sticking one of them in your safety deposit box, locked desk drawer at work or another secured space is a good backup.

Personally, I don't, since I've never wanted to log into my Google account on a device I encountered while out of the house. I'm not really sure why you'd ever do that IMO.