Hi Colin!
You'll probably get a different answer to that from every one you ask! Few things .. different varieties have different needs eg. 1.koi will grow rapidly & to quite a large size, so you'd be better off to under-stock to start with & 2.they'd eat substancially more doing so & 3.therefore produce more 'waste' & possibly need more work. 4.More 'waste' could also equate to more algae blooms in hotter weather. Here's a few more 'bits' :
The amount of fish you can stock will depend on the capacity of your pond and the adult sizes of the fish you want to introduce. Remember to reduce water quantity for the displaced amounts caused by potted plants etc in the water.
One of the easiest ways to get an idea of how many fish your pond can accommodate is by using a Rule of Thumb of 1" per gallon. Sorry, you'll have to do the math. Then you have to consider this .. fast swimming fish eg. koi, commets, etc need more water than slower swimming breeds eg. fantails, ryunkins etc. Rule 'of thumb' here is 20 gallons per adult fast swimmer & about 15 gallons per adult slower swimmers. I think this would be 1cm of fish per 1.5litre to 2 litres depending on the water surface area. The deeper the pond the less fish per litre. If you have a filtration system in place, this should be ok .. but remeber bigger fish need more space.
Larger bodied species, such as goldfish, displace more water and produce more waste than a smaller bodied species of fish.
When you perform your 'rule of thumb calculation', base it on the approx. adult size of the fish. If you intend buying mature fish, measure the fish from its head to the base of it's tail. The fish in the pond pic I posted, was only about 1.5 inches long when I bought him as a 'yearling' 5yrs ago .. they grow VERY fast in large bodies of water! One of biggest boo-boo's people make is putting too many fish in a pond too soon .. they WILL breed all by themselves from a very young age & small size .. you'll soon have more at NO cost!
Even though they will eat their own eggs during spawning, you'd be surprised on how many babies you'll find later & think 'where did THAT come from?' My previous pond had no introduced fish in it .. only plants I'd bought from a local grower .. low & behold, there were baby fish in there & I hadn't even bought any! Eggs had been attached to the plants!
To tell you the truth, I don't know EXACTLY how many I have in my bigger pond .. I catch some of the smaller younger ones for local sale & there always seems more to replace them .. the bigger ones form the 'stable' population & will for some time if they reach their expected old ages of 10-20yrs old! Geez, I'll be on an old age pension by then!

DON'T be tempted to add those little fish to your pond some people call 'guppies' with black spots on their flanks .. they are 'mosquito fish' & will pull the tails & fins apart on goldfish! Horrid little things! Good for yabby bait! Here's a few more pics .. tempted them no probs to pose for the pics! Cheers for now, Dianne.

ps. would a link to a video I did work here on a forum page & is it allowed? I have a vid of my fish swimming in the pond .. no Academy Award stuff, my first & only fish vid effort!