I've been watching the midseason playoffs for the LCS the last couple days and one of the commentators today made an offhand comment insulting one of the players in the game. Genja has a habit of building unorthodox items and people love to rag on him about it. In this case it wasn't even that he'd built anything weird in this game, as it looked pretty standard to me. No, he just wanted to insult Genja for some reason. At any rate, he was bringing up a game in the past where Genja declined to build the item last whisper and he said that not bringing the percentage armour penetration from that item was going to give the opponent three, four, or even five times as much effective health. That seemed like a patently absurd claim, but I wanted to check it out. Especially since I sometimes get yelled at in ranked games when I don't build an early last whisper. I get told I don't do any damage with just bloodthirster, infinity edge, and phantom dancer against a team without a lot of armour. I don't like getting yelled at. Why must people yell?
Anyway, there's actually a lot of variables at play when working out just how good last whisper is going to be. Some champions like Corki, Ezreal, and Varus do a reasonable amount of magic damage which will reduce the power of armour penetration. Some enemy teams won't build much armour. Maybe you've got teammates who have armour shreds themselves to help you out? I'm going to try to simplify all of that and just assume all of your damage is physical and your team isn't going to help you at all. I will vary the enemy armour, especially because I want to see just how much armour it takes that building a non last whisper item will quintuple their effective health.
There's also the question of where it fits in your build order, or what you'd have to replace it. I'm going to assert you'll be building berserker greaves and bloodthirster first and will be adding in phantom dancer and infinity edge after that. Looking at a straight auto attacker like Caitlyn (who actually has 2 magic damage abilities, but they're not really used for damage) where are we standing? I'll assume she's using max rank peacemaker on cooldown and otherwise not doing damage with anything but auto attacks and that she's level 15 for the AD and attack speed gains per level. AD quints and armor pen reds will be the damage related runes, and pretty much all of the damage masteries are in play.
I built a spreadsheet to compare the different item options and it was actually pretty interesting. At an enemy armour value of 100 if you're only allowed one more item the most efficient item is last whisper. But if you're allowed two more items you actually want infinity edge and phantom dancer. The way those two items compound with each other pushes them over the top of last whisper with only one of them. But in both cases it was all pretty close. We're talking a difference of 5% or 7% in terms of damage done, which is small enough that I don't think anyone should be yelling about one choice over the other, especially since the 'right' choice varies depending on how long the game gets to go.
How about if the enemy has more armour? 200, say? Well, then last whisper is the big winner. It's 33% better as a single item and 7% better when combined with the phantom dancer.
What about less armour? The two people you're most likely to be fighting in the mid game are the people in your own lane and neither of them are too likely to build any armour. They might have, say, 60 armour? In that case last whisper is the worst single item and should be kept out of the duo items. Starting phantom dancer is 16% better and adding on infinity edge is 19% better.
So it seems pretty clear to me. If you'll be hitting someone without any armour items you don't want last whisper early. If they have a fed tank who built a ton of armour and will be in your face you need last whisper as soon as possible. If neither is really the case then it doesn't much matter. It's probably 'right' to build it first, but it depends on how much fighting you'll do at the one extra item mark compared to the two extra item mark. Oh, and while it's the most gold efficient item against the 200 armour people it's actually still the worst actual damage item. It just gets to win because of how comparatively cheap it is.
Now what about the ludicrous statement that skipping last whisper will make the enemy take FIVE TIMES as long to kill? How much armour do I need to give the enemy to make BT+LW+IE+Doran's Blade do five times as much DPS as BT+IE+PD? (The gold difference is 500, so I'll add in an item worth about that much to even things up.)
There is no such number. Even with 10^300 armour I can only make the last whisper set do 16% more damage. Maybe doran's blade is that abysmally bad? Or maybe I should be using PD instead of IE? It was the better set at 200 armour, though it has an even bigger gap of gold. I guess I can balance out that 1500 gold with an extra BF sword? That's pretty fair. So comparing BT+LW+PD+BF against BT+IE+PD I end up with the first set doing 4% more damage at 200 armour, and at ludicrous numbers of armour it still only ends up at 24% more damage.
Maybe the commentators are thinking Genja took the 2300 gold he could have spent on a last whisper and instead spent it on hookers and blow? Maybe if we just give out a free last whisper we can reach five times damage done? So let's compare BT+LW+IE+PD with just BT+IE+PD. In this case the first option does 52% more damage at 200 armour and 81% more damage at 80 million armour.
So getting a free last whisper is pretty good! But it doesn't cause you to do double damage. It certainly doesn't cause you to do quintuple damage.