This page starts off with a list of games you might try. Then it ends with some outdoor game videos from my YouTube channel.
Monkey Chase You need a large playing area with a centre-line. On each side of the centre-line, have a circle inside of which is a monkey (perhaps 25m back from the line). Also have a jail (perhaps 25m back from the line and 50m away from the monkey). Split into two teams and take one side each. You win if you grab the monkey from the other side and get it to the monkey circle on your side. You are safe if you are in your half. You are also safe in the monkey circle on the other side. If you are carrying the enemy monkey then you are not safe in your half. If you are tagged you must go to the jail on the opposing side. If you are tagged with the monkey immediately drop it. If you get to the jail without being tagged you have released everyone in the jail.
A good strategy is to keep probing and wait for the other team to make a mistake by not covering your route to the monkey or some other strategic error. It is typical for the game to start slowly but eventually a rash move will result in a dramatic win or a sudden advantage for the other team as you go to jail. If you make it to the enemy monkey circle you are often stuck there for a while until you see a chance to run for it: I have sometimes been stuck for about ten minutes. Note, if you are in the enemy monkey circle and the other team are about to run the monkey in then you can just jump out and tag them at the last second to prevent them from winning.
Dodgeball There are lots of variants of this game.
One variation is to play with a big circle (perhaps 20m diameter) and have the throwers on the outside. If you are hit then you join the throwers outside the circle. Winner is the last one remaining inside.
Another variation is Run-the-Gauntlet Dodgeball. Everyone takes turns running down a corridor, perhaps marked with cones on a field and maybe 5m wide and 40m long. One or more people start the game as throwers and try to hit you as you run down. If you are hit you become a thrower. The winner is the last person left. You could then give them the chance for a last glory run avoiding everyone. A good way to manage this is to allow one ball per two throwers (so obviously start with two people throwing at the start): so as you are hit you go outside the passage and wait for a second person to work with standing on each side of the corridor throwing the ball to each other.
If you are playing the more standard version of dodgeball you could play the slight variant called David and Goliath: One player is Goliath and if they are hit then the team automatically loses.
Fugitive One player is loose on the pitch! The other teams each have a ball and compete to hit the fugitive (I've seen it work with two teams of 6). Team members can't run with the ball which is important as it gives the fugitive a fighting chance to survive for a while and get some glory. You could time everyone and have an individual prize for whomever lasts the longest, however my experience is that it works best if a good runner always plays the fugitive and just enjoys the fun of it as many people find the running too difficult.
Zombie Have lots of bits of paper in a hat: the same number as number of players with (for example) two having a Z on them and the rest have an H. Everyone takes a piece of paper at random and secretly looks to discover if they are a Zombie or a Human. Everyone now starts the game knowing there has been a zombie outbreak and hoping to survive. The zombies start playing as humans and can choose an opportune moment to turn into a zombie (for example when they are hiding with someone in a cupboard)! Once you are a zombie you have to do some acting and slowly walk around acting stupidly with arms outstretched perhaps moaning or wailing or saying "brains". If you started as a Human then as soon as you are tagged by a Zombie you also become a Zombie. The last person left wins but will also eventually be infected to add to the fun! This can work well on a marked-out area on a field or in a building.
Races with a Difference Instead of running 100 metres on your own there are various ways to make it more awkward and fun. Obviously be careful in case of accidents.
Cage Race Two (or more) people hold hands (facing inwards) around a central person to create a team. Your team has completed the race when the central person crosses the finish line. A referee might watch and send you back to the start if you break the rule of the race.
Hands on Ankles Run on your own but have your hands holding your ankles the whole time.
Rhubarb Race Move in threes with your backs facing and hands up in the air touching at the top.
Crab Race Position yourself in pairs with elbows hooked while back-to-back. You may well find it easier to travel sideways.
Centipede Race Travel on all fours, holding the waist of the person in front. Centipedes can be long but keep the race short as it can hurt your knees.
Space Hopper Assault Course Set up an obstacle course. Get on a space hopper. Time each other or if you have enough space hoppers everyone can go at once or maybe have heats (although I'm not keen on heats as I think it is less fun for people who are out early).
Circle Hunt Teams of people stand holding hands (facing inwards) with one person in the middle. The teams compete to see who can collect the most tokens scattered around the play area. The tokens must be picked up by the person in the middle. Obviously be careful about going too fast and having a big pile up.
Capture The Monkey Nerf Game (like Capture the Flag) (YouTube)
Rescue the Prisoners; a Nerf Game (YouTube)
Dodgeball played with Giant Balls (YouTube)
A Fun Game of Hockey with Plastic Sticks (YouTube)