My OS disk uses LVM and is encrypted. I've recently had an issue with it and want to reinstall Ubuntu without formatting my /home.
In the installer partitioner it shows 2x partitions, a 767MB partition, which is /boot and the rest.
If I mark the 767 MB partition as /boot and try to install it says no root filesystem is defined. If I mark it as / it says that the root partition needs to be at least 6.2GB.
How do I get around this?

Edit:
There doesn't seem to be any option in the partitioner to decrypt anything so I did so in Gnome.
I assigned /boot to the 767MB partition and / to /dev/mapper/vgubuntu-root which I didn't format as it has a size the size of the drive.
Install seemed to go OK but on booting to it I got dropped in a busybox shell. What causes this?
Is there a way to safely format / without risking my data?

I guess the /boot partition is something that became necessary with UEFI firmware. Some people call it (UEFI) the BIOS, but that isn't the official name. BIOS is the old system and UEFI is something very different, build differently from the ground up.

The /boot partition tends to be 100 MB from what I have seen. I don't know why yours is so large.
Don't delete it. Don't format it.

You need to create another volume and label it as / which is the equivalent of C:\. You probably want this to occupy most of your HDD. Format it as ext4 or whatever your favorite is.
You need to create another partition and label it as the swap. Maybe make this 100 GB. I think you can even get away with 10 GB if you have plenty of RAM and if you think you don't really need swap.

I doubt very much the OP needs to create any partitions. One would imagine the prior install had all needed. Pity it appears there isn't already a separate /home, that would make life easier.Because once you opened the encrypted container and mounted the lv's, the installer just saw those lv's and wouldn't know to include the LUKS (presumably) and LVM support.Not if you don't have a sparate /home partition or lv. That's why backups were invented.