The
Skype website says that UK "Toll Free" numbers cost 0ppm, with no connection charge. UK "freephone" numbers begin 0800 and 0808, with a few still in use on the 0500 prefix.
It is possible to call freephone numbers from inclusive minutes on your mobile using a 01, 02 or 03 dial-through number. A couple of examples are
0800Buster and 020 0222 0900. Once the call answers, enter the destination number (but make sure that you
don't press the Call button after doing this else it will set-up the call via your provider).
The range numbers 0808 80x are free to call from the main mobile providers, including 3. This is via an
arrangement with the Helplines Association, but only certain helplines qualify for such a number. For example, NSPCC has 0808 800 5000.
Calls to 0845 numbers made via Skype Pay As You Go United Kingdom cost a one-off 6.9p charge plus 9.2ppm. Durations are rounded up to the next full minute, so any part-minute will be charged as a full minute.
18185.co.uk is a service that you register with. It is a dial-though provider which is accessible via 020 and 0808 numbers. The call charges via these numbers are published
here. A call to a 0845 number will cost 5p plus 3ppm at all times (as with Skype, for billing purposes, durations are rounded up to the next whole minute). Obviously, you will be billed by your phone operator as well (if applicable) for the call to the 020 or 0808 number, and this will be until you hang up (i.e. at the end of the call to the 0845 number).
With 18185 you tell it which number(s) you call from. If your Skype account has a UK phone number (one which it presents on the recipient's caller display, which can
apparently be your mobile number), then you might be able to use Skype to call via the 0808 number. Or you could call via the 020 number (with calls to the 020 coming from your minutes). With both these possibilities, 18185 will charge 5p plus 3ppm for 0845 calls.
Sipgate charges 3ppm to call 0845 numbers. It says that there are no minimum charges and calls are "billed on a per minute basis". It doesn't seem to say whether it rounds durations up to the next whole minute which, if it does, means that the minimum cost of a call is the first minute. Either way, it is still the cheapest way of calling 0845 numbers out of those providers I've looked at here. And the prices for 0844 and 0871 are also a bit cheaper than BT's, and this is going some as you can't normally beat BT's due to regulation on it which forces it to charge a very low price for these ranges. ("Very low price" is in relation to the cost of these ranges only, and not relative to calls in general.)
giffgaff offers free calls to freephone numbers.
I've never used Sipgate and I don't have any interest in it, but it does seem to offer very favourable prices.
There are people on here who are far more knowledgeable than I when it comes to finding the cheapest rates, so I put the challenge to them to better what I've suggested.