Let's face it: if you don't have Rich Parents (or Traits in your game at all), you're going to try and make every copper stretch when gearing up. By a train of Alchemic thought along the lines of, "Wow, longbows are expensive. I haven't taken Weapon Focus on anything, so let's see what alternatives I have until I get Explosive Missile. Oh, slings shoot stuff. I don't think a bomb will be able to fit, though... why not just make a larger sling?" we get to this thread.
So, let's start by looking at the sling, as sized appropriately:
d20pfsrd's take on the Sling:
A sling is little more than a leather cup attached to a pair of strings.
Description: Your Strength modifier applies to damage rolls when you use a sling, just as it does for thrown weapons. You can fire, but not load, a sling with one hand.
Action: Loading a sling is a move action that requires two hands and provokes attacks of opportunity.
You can hurl ordinary stones with a sling, but stones are not as dense or as round as bullets. Thus, such an attack deals damage as if the weapon were designed for a creature one size category smaller than you and you take a –1 penalty on attack rolls.
With a small leap of faith, we can say a sling is a one-handed weapon but requires two hands to load, akin to a hand crossbow. The rest of what I say operates upon this fact: that even though it is under the ranged weapon category, the description given would allow one to call it a one-handed ranged weapon. With that out of the way, let's continue with the PRD's weapon size rules:
PRD's CRB Equipment:
A weapon's size category isn't the same as its size as an object. Instead, a weapon's size category is keyed to the size of the intended wielder. In general, a light weapon is an object two size categories smaller than the wielder, a one-handed weapon is an object one size category smaller than the wielder, and a two-handed weapon is an object of the same size category as the wielder.
Inappropriately Sized Weapons: A creature can't make optimum use of a weapon that isn't properly sized for it. A cumulative –2 penalty applies on attack rolls for each size category of difference between the size of its intended wielder and the size of its actual wielder. If the creature isn't proficient with the weapon, a –4 nonproficiency penalty also applies.
The measure of how much effort it takes to use a weapon (whether the weapon is designated as a light, one-handed, or two-handed weapon for a particular wielder) is altered by one step for each size category of difference between the wielder's size and the size of the creature for which the weapon was designed. For example, a Small creature would wield a Medium one-handed weapon as a two-handed weapon. If a weapon's designation would be changed to something other than light, one-handed, or two-handed by this alteration, the creature can't wield the weapon at all.
So we get to a sling being a two-handed weapon if we increased it's size category. Except the action of loading an appropriately sized sling is already a two-handed action: so would you be able to load a large sling by yourself at all? If it follows the same rules as the rest of the weapon I would imagine not. Sadly, it only gets into more grey area from here-on out: would a large sling give you one and a half your Strength mod for projectiles since it requires two hands to use? Would it be large enough to use with potions, alchemical weapons, or the like? (I'm assuming the typical flask is larger than a common stone, which prompted this idea in the first place.) Even better, would a large sling be a two-handed melee weapon for those characters that took the appropriate feats?
TL;DR:
A sling requires one hand to use, two to load, like a hand crossbow. Which makes it a 1h weapon. A large sling is a 2h weapon, but gets into a lot of grey area really fast, specifically:
Can it be reloaded at all?
Do you get 1.5* your Strength mod?
Does it become a two-handed melee weapon if you qualify?
Could it be used to lob alchemical weapons, etc. since they're larger than a stone?
I'm looking for how everyone would interpret this, with evidence if possible. (And because I know someone is going to bring it up eventually, the Bomb Launcher just doesn't fit the same vein since it only augments your bombs, where a large sling is much more versatile)
The classification of effort: light, one-handed and two-handed, is a statistic that applies to melter weapons only. It does not apply to ranged weapons, regardless of how many hands are employed in operating them.
I think one thing that may help is to take a look at the 'Ammo Drop' feat from the Halflings of Golarion. PFSRD link: http://www.d20pfsrd.com/feats/combat-feats/ammo-drop-combat
Spoiler:
Your coordination is so perfect that you can simply drop ammunition from your hand at the exact moment required for it to fall into an open sling as your twirl it around.
Prerequisites: Sleight of Hand 1 rank, proficient with sling.
Benefit: You can load a sling or one end of a double sling with one hand as a swift action or move action. This does not provoke an attack of opportunity.
If I'm reading this right, that means you can load a normal sized sling with one hand when using this feat. If you had a large sling and this feat, I think you'd be fine. I think.
As far as the strength mod goes, no, I don't think using a large-sized sling would give you 1 1/2 times damage. However, I can't point to any piece of evidence and say "This proves my point." The closest I can find is the feat Two-Handed thrower.
Spoiler:
You hurl weapons with both hands and with great force, sometimes using a whirling technique to send your weapon flying through the air at tremendous speeds.
Prerequisite: Str 15.
Benefit: Whenever you use two hands to throw a one-handed or two-handed weapon, you gain a bonus on damage rolls equal to 1-1/2 times your Strength bonus. Using two hands to throw any weapon requires only a standard action for you. If you also have the Quick Draw feat, you can throw two-handed weapons at your full normal rate of attacks.
Normal: You add your Strength bonus on thrown weapon damage, regardless of available hands. Throwing a two-handed weapon is a full-round action.
Note the 'Normal' section - Regardless of available hands, you normally cannot get 1.5xSTR damage. You get 1.0.
If you want to wield your gigantic sling as a melee weapon, I'd just get the feat that lets you wield a sling as a melee weapon. It's called Sling Flail. ( http://www.d20pfsrd.com/feats/combat-feats/sling-flail-combat ) In this case, you'd be wielding your large-sized sling as a large-sized flail, i.e. a two-handed melee weapon for a medium-sized creature. Kudos. You'd still take a -2 penalty for the sling / flail being designed for a larger creature, I think...
Finally, I think you could probably use it to lob non-stone projectiles. I might impose a penalty to that, if I were the DM, such as a -2 penalty (on top of the -2 penalty for it being a big freaking sling) since the weapon wasn't designed for that, especially when an item like the Flask Thrower exists specifically to lob alchemical substances farther.
While ranged weapons (other than firearms) don't work from the standard "handiness" categories, there is a general principal of equivalency. You can't say that, since there's no "effort to wield" on a Bow, there's nothing to step up from size increases and say you'll wield a Colossal Bow 4 sizes too big for you. But, since the exact equivalencies aren't always outlined, you have to apply the basic principals of "effort to wield" to determine which category a particular weapon best fits into. It's also worth noting that the rules state that, unless the weapon specifies otherwise, to treat ranged weapons as requiring two hands. This sets the de facto standard for projectile weapons (other than firearms) as two-handed weapons.
For the Longbow/Shortbow and Heavy Crossbow/Light Crossbow, it's pretty obvious that they are equivalent to two-handed weapons. The Crossbows explicitly states that it takes two hands to wield normally, but can be wielded in one hand with a penalty (special exception, still would be treated as two-handed weapon) and special considerations for what TWF penalties to apply when using one in your off-hand (again, specific exceptions). The Bows state that they necessitate two hands to wield no matter the size. So, even if you use a Small Longbow and count it as a one-handed weapon (or a tiny one as a light), it's one with which you are obligated to commit two hands anyway.
Hand Crossbows and Slings, on the other hand, explicitly state that they can be wielded in only one hand. If you consider them as one-handed weapons, it stands to reason that they could be scaled up into two-handed weapons if you use a Large variant. A two-handed Hand Crossbow would be a comically large crossbow strapped to your hand which you must support using the other hand to fire properly (it's still hand operated as opposed to lever-operated like the light crossbow). The Sling would just be a larger Sling that hurls bigger stones, but must be swung around two-handed. Presumably, you'd still be able to "hold" it in one hand while loading it with the other since handiness categories are how hard they are to attack with; you can always "hold" a weapon that's too big, provided you can handle the raw weight, which is, essentially, what you do when you reload it; you're "holding" the weapon, not "wielding" it.
However, since these are just equivalencies and not actually the categories used for melee weapons, you wouldn't get 1.5x damage since the rule that gives that explicitly calls out melee combat. So you wouldn't get 1.5x Str to damage neither for a Medium Sling used with two hands, a Large Sling which must be used in two hands, or a Greatsword thrown as an "improvised thrown weapon" (which takes a full-round action).
