According to gov.uk my invoice must include "the company name and address of the customer you’re invoicing". However my customer does not have a company, I'm simply invoicing them as an individual. Do I still need to include their address?
Yes. Even if you invoice a private person (no company involved), then the invoiced person & their address who pays bill must be named on your invoice. There are two tax and one criminal reason. a) in your accounts you are showing a true record of source of payment to you. b) This keep your tax accounts in order. c) If "No person" & "no company" is shown on your invoice, the source of the funds is unclear and you could be into tax evasion or obtaining funds from criminal sources. This gives you a big problem with both HMRC and police if it comes into an investigation.
Cash receipts in a shop are not invoices but tally cash in with goods out to unnamed persons. That is why cash economy is truly "untraceable".
In a Danish context, the client address is required as the inclusion (or not) of sales tax on the invoice depends on location (among other factors).
Domestic clients must pay an additional sales tax, while foreign businesses may be invoiced without added sales tax.
Rules in your jurisdiction may differ. In addition, I could imagine the authorities could extend the thinking behind e.g. banks having to 'know their clients' to your business.
For this reason I would always include an address - as it seems the authorities seem to prefer/demand a paper trail of where money originates from and where it ends. Not being able or willing to explain who or where your clients are, could be considered shady and cause needless problems.