When jungling in League of Legends, you could kill the little minions first, so their damage goes away, but then the big guy is beating on you longer, or you can kill the big guy (presumably the greatest source of damage) first to shorten that punishment.
What is the optimal killing order within each jungle minion pack to minimize damage taken? Does it vary from group to group?
I would say 99% of the time you should go after the largest minion first. Especially in the case of the minions with buffs you should always aquire the buff, which will probably in turn make killing the other minions a bit easier. The killing order overall and the groups you should go after first is based on your skills and your champion, but after your first run through all of the camps you shouldn't be worrying about taking the beating. Also,if you're concerned with taking too much damage ask a teammate to leash the big guy for you.
As a rule of thumb, you should always go for the larger minions first. Early game, they're going to be the ones that cause significant chunks of damage to you (much moreso than the smaller minions). Since the recent jungle revamp it's even wiser to do so since killing the large minion at small camps (read: not Blue/Red) rewards you with a small health restoration that might save you when you may have otherwise died. Additionally, the large minions have the bigger money bank, which is important to know when counter jungling. Stealing any of the large minions while leaving the small ones can have a significant monetary impact on the enemy jungler early game.
However, that being said, there are merits to killing the smaller minions first. If you're aware of the timers for the camp and you know that the large minion is about to roll over to a higher "banked" amount for gold/xp, it may be wise to kill the smaller minions first so that the larger one can accumulate a higher value before you kill it. For certain characters and abilities like Nunu's Consume, Gangplank's Parrrrley, or Trundle's passive, it may be easier to kill a smaller target for the effects of the ability during heated moments.
I disagree that it should be the largest minions first really. Only in the early game, where your defence is very little, and red/blue can deal a lot of damage to you, should you probably aim for the bigger one first, however, depending on your champion, if you're using a champion that is very good at, or designed for jungling, then you could probably survive the damage that the big minion does to you, whilst you lessen the total damage by attacking the others.
Assuming you won't get ganked, and your champion isn't squishy, or has a sustain, it's almost always better to take the little ones first, because if you want to get the red or blue buff, you don't want to waste it's precious time, by fighting the minions which you don't need. You'll be able to utilize the maximum time you have the buff for, if the lesser minion are already gone.
Even when facing super minions in the lanes, getting the lesser minions means more money and exp, then you can focus on the big one, otherwise your minions would kill the majority of the lesser minions, thus wasting all that nice gold and exp.
From mid game and onward, unless you think someone else will try steal your buff, I highly recommend that 99% of the time you attack the lesser minions first. Especially with champions like Warwick and Xin Zhao, who have sustain abilities.
If you're unsure if you'll be able to kill the camp, go for the largest creep first.
At spawn, the ancient golem does 65 damage vs the young lizard's 11 (the lizards have slightly more attack speed). Since armor applies to each shot, the golem does significantly more damage over the course of the fight.
Later in the game, it can be advantageous to kill the lizards first and then lure the golem into the bush. This allows you to hide while fighting it, and gives you a chance to run away should enemy champions show up.
edit: Kha'Zix gets a bonus when he isolates a target, meaning he should always kill the small minions first.
