Does anyone offer hosting of PostgreSQL databases, with PHP, for no
more than 5 UK pounds per month? (That's around $8.80 US.)

I'm not interested in those free ones where you have to post large
numbers of messages in their forums first.

There are many reliable PostGRESQL providers. The following is an
indicative list a little above the price points mentioned.

http://www.oxxus.net/hosting_database.htm (though 10$)
http://www.eroute.net/index-en.html (10$)

These plans have not setup fee, which means they will effectively
touch your price  requirements over a year.

Go lower than this price and u will get just junk sites.

Easier still connect to a DSL or cable line and run your own site from
home. If your provider switches IPs every few days then use dynamic
DNS services to keep track.

Srikanth