since it is possible to hit multiple targets with the same action, would it be possible to hit the same target multiple times with a single melee weapon?

No !

Hough !
Medicineman

I'm trying to think of loopholes, but it seems that the only RAW way to split a dice pool to perform multiple attacks on a single target is to dual-wield. There are ways to do the opposite (use a non-split dice pool to hit multiple nearby enemies), but I can't find anything to suggest you could use a single weapon to hit the same guy twice.

At least with dual wielding the system makes sense - you swing two katanas at a guy, you hit him with two different weapons and apply damage from each. A single melee weapon just doesn't seem feasible - you swing your weapon at the opponent and hit him, then somehow manage to swing again for another hit? You'd need to have incredible reflexes to swing that fast - and if you do have those reflexes, we call that having extra initiative passes.

Now, I would definately argue that it should be possible to do what you're trying to do if you use Unarmed Combat, because there's nothing stopping you from using two "weapons" at the same time in that situation (with off-hand modifiers, naturally). The only argument against that would be the fluff about Unarmed Combat being more complex than just throwing a single punch per attack (SR4, P. 146), but that same fluff also applies to Melee Weapons, and you can dual-wield those just fine.

~Umidori

Some fraghead is going to take this to the logical extreme and make up an attack including a headbutt, 2 punches, 2 kicks, 2 knees, and a bite, making a grand total of 8 unarmed melee attacks.

Hey, wait...

Attack of the double post!

And it can only get worse if they have more than 2 arms and 2 legs.

A melee attack is representative of a whole action phase worth of attempted assault using whatever is at your immediate disposal. The split dice pool rules for multiple targets and mutliple weapons are there to reflect spreading your violent attention thinner. So whilst attacking the same guy twice with the same complex action is covered by the one attack roll, you could theoretically reason the use of the multiple targets rule and split your pool.

Issue 1: modifiers would count double. To account for the difference between two targets, modifiers must be applied after the split, so if you split your pool against the same target with a +1 reach bonus, you would get +1 to each pool and thereby con an extra die out of the system.

Issue 2: the defender gets a defence roll and a damage resistance roll per attack, making a split dice pool (all things being equal) much less effective than a whole one. He does lose a die from each subsequent defence roll for receiving multiple attacks, but this will still be much better off than splitting and his damage resistance will be at full whack for each attack that does manage to get through.

There is enormous scope for abusing the split dice pool rules against a single target, especially since you'll never do this unless it's to your advantage (assuming you have a good idea of the opposing DP).

It could be argued that using two fists should use the same mechanics for using two weapons, though I would argue that two fists are de facto one single weapon in melee circumstances. Otherwise you have the GM asking whether a PC is attacking with both hands or just his on-hand with the PC then weighing up the pros and cons of leaving one half of his weaponry hanging by his side.

You could, perhaps, create a 'berserking' house rule for situations where the character wants to overwhelm the target with a flurry of less co-ordinated attacks.
Player: I'll berserk the guy
GM: How many attacks?
Player: Um, I think three on this occasion
GM: Ok, your DP of 10 is now 4, 3 and 3. He has a DP of 10, then 9, then 8. Good luck.

Ultimately, the situations whereby multiple attacks on the same target helps is a good idea will be few and far between and taking advantage on these occasions is pretty low even for powergaming.
IMO, RAW works fine.

It's what is already happening, you are just using one die roll to do it.

Otherwise you can just use more IP.