Well, I am impressed! A lot of y'all got one or two right but the one that got closest to naming them all is Mr. Maloy...
The round on the right is indeed a Remington Rimfire Magnum. The round was designed by Remington and went into production in 1969. It was Remington's answer to the .22 WMR although the round never took off as planned. It was a bottle neck design and was more expensive then the .22 magnum. The other problem was being a bottle neck cartridge many shooters felt it should be reloadable to save money. Remington produced two rifles, Remington model 591 and 592, to shoot the new cartridge. The rifles were both bolt actions with the only difference being one used a tube magazine and the other used a box magazine. There was also a run of barrels in the new caliber for the Thompson Center. The failure of the new cartridge to be accepted resulted in the rifles being taken out of production in 1973. Remington continued to produce the ammunition in the new caliber until 1981 at which time the round was labeled obsolete and production ceased. In 2008 Aguila took up production of the cartridge and manufactures it today. Both of the Remington Rifles have become collectable and bring prices in the $500 to $600 range.
The second round from them right is the .22 TCM of Rock Island Armory fame. Not much to say about it that hasn't been said except that is a fun round to shoot and is available in both handgun and rifle variants.
The third from the right is indeed the .22 Winchester Automatic Rimfire for the Winchester 1903 rifle. It is not a .22 LR and is actually only to be used in a Winchester Model 1903 rifle.
Finally, the little guy on the left. While it might be called a .22 short it is the history of the round I was trying to bring out. The round is a Remington Stun Safe frangible for use in a Remington Slaughter House Rifle. It was particularly developed for cattle, hogs, and sheep.
Thanks for taking part in my little fun exercise and if anyone wants more info just let me know....
Oh, and in case anyone is wondering, I have all of these firearms in my collection!
Don