How much are you willing spend? LOL. Some people said they could using SSD drive.. so it could be a few TB in size if you want to spend the money?
Actually, probably not. FAT32, which is the only filesystem I know works with TeslaCam, has a 2TiB filesystem size limit. Thus, anything bigger than that will be wasted money. That said, perhaps it will work with some other filesystem. Personally, I tried it with ext4fs and it did not work, so that's out of the question, unless I erred in some way. Others have confirmed it does not work with exFAT. I doubt if it would work with NTFS or HFS+. Those are all the most likely filesystems, but maybe it'll work with something more obscure.
I deleted good amount of videos and plugged the stick back. Started recording like normal. Drove somewhere and activated sentry. When I came back, it was again grayed out. Darn! Tried unplugging etc but no help. Formatted the stick again and now it's been working since yesterday.
I've discovered that Tesla's computer does not unmount the filesystem when you stop recording. Thus, when you stop recording and pull the USB drive out, you're leaving the filesystem in an unsafe state, which is likely to cause filesystem corruption. You should always perform a filesystem check/repair on the filesystem when it's in such a state. Chances are that such an operation would have fixed the problem and gotten it working again for you, although I can't be 100% sure of that. (It could be it was so badly damaged in your case that a filesystem repair wouldn't have worked; or it could be that your drive is physically failing and creating a fresh filesystem just "plastered over" that problem.) I've found that the same is true of music media, even though those should probably be mounted read-only to avoid this problem (unless Tesla is storing file-playback position data or something on the music media).
Tesla really should provide a "safe eject" type option. Used properly, that would most likely eliminate these problems, or at least greatly reduce their frequency.