The katana is an aesthetically pleasing piece of craftsmanship, and certainly an effective killing tool for its time and environment. The thing is, people go overboard with the hype. What is the katana? Here's what: a sword. Specifically, a one-and-a-half-handed, curved single-edged longsword. That's it. Its main feature is that it's an impressive feat to make something that good with the poor steel they had to work with, and even more impressive to make it that pretty, but while I don't necessarily think you can't put it on number one... I'd say you'd better say "1: longsword" then.
Personally, I feel the best way to rank weapons is by how much it would shorten the time it would take to train an average person of its time into an equally or more competent killer than a wielder of its predecessor. I'm not saying that the wielder of one weapon would necessarily defeat the wielder of another, but rather ask how long would it take him to be as effective at killing an archetypical opponent with this new weapon. In lesser terms, factors such as encumbrance, versatility and such would come into play, though they can be hugely important as well. As such, listing what place they should all be is fairly pointless to me, but these are the contenders, in my opinion, in roughly ascending order:
- honorable mentions go to: all kinds of modern and semi-modern military weaponry such as machine guns, artillery, bombs and such. These are not, in my opinion, "improvements" over older weapons, but rather answers to specific tactical challenges, a "custom tool", but you can't fully ignore them. Also combat vehicles, be they land, sea or air vehicles. These are usually not weapons, but rather weapons platforms. Halberds, because while there was little significantly new to them, they combined them marvellously in an impressive troop weapon. And miniguns, for being to firearms what katanas are to melee weapons - damn cool but not all that special.
- handguns (sadly replacing knives). I dislike having to put these here. The sad fact is, modern handguns are too easy to use and too readily available, taking nearly all the effort out of killing someone - a handgun is comparable in encumbrance to a knife, many are easily concealable, but it takes very little practice to be lethal in combat with one, and as you add slightly more practice you can probably stand up to any other entry on this list provided you are within the handgun's effective range. Sure, a knife is deadly, but a handgun beats it by so much it's not even funny.
- rifle (replacing leadball longarms). The jump in accuracy, power and effective range so vastly outstrips any previous technological jump that it easily punishes leadball longarms in this list - the leadball weapons, blunderbusses, and cannons that came before it were in many ways better than bows and crossbows, but by a far smaller margin than the one with which rifles beat them, in turn.
- bow (replacing a plethora of thrown implements and the like). Ranged combat before the bow was very limited in range, accuracy, or both. It's much easier to be accurate with a bow than it is with javelins, rocks, slings, and all manner of such projectiles.
- crossbow (partially replacing bow). A skilled longbowman may be more than a match for a skilled crossbowman, sure. But developing and maintaining a bow arm and aim is FAR more difficult and time consuming than learning to look down this length of wood and depressing this mechanism to kill whatever is in the axis that extends from it.
- sword (replacing axe where it's affordable). More manoeuvres that can easily kill, harder to miss outright, a soldier's favourite any day over the axe. The axe still has its uses of course, but hey, there are surprisingly few things an axe can do that a sword cannot.
- axe (replacing and complementing spear). A bit harder to make than both spear and club, it's nonetheless a much more effective killing tool, and any recruit would appreciate how much easier an axe kills your enemy than a club, and the very straightforward way of wielding it. Indeed, most recruits would have wielded one prior to enlisting, since they'd more than likely spent time working with one.
- spear/sharp stick of various designs (replacing club). Over its predecessor, a huge increase in ease of use allows you to wield it for far longer, carry it further, and while it requires some tools to make one it's not a terribly complicated or difficult process.
- club (replacing sharp rock). More reach and easier to wield than the sharp rock and, depending on design, equally or more effective.
- sharp rock (replacing unarmed body). You have any idea how much practice it takes to reliably kill someone with your body in combat? Picking up a sorta sharp rock will let you match the effectiveness of a hardened unarmed fighter in a far shorter time than said unarmed fighter spent on his craft.
- knife (special case). Knives are harder to make than sharp rocks, for sure, but still they have been fairly easy to get a hold of for a lot of cultures and in lots of time periods. What they would replace was in general being unarmed in a situation where due to social norms or surprise you were not armed, and the deadliness of the simple knife far surpasses most readily available replacements.
- unarmed body. It is a remarkable tool for creating things, sure, but a well-wielded human body is a deadly weapon for sure, capable of incredibly versatile styles and feats of combat. The best part, though, is the procurement cost - we each come equipped with one from the get-go. Plus, as far as I'm concerned, it's the true minimum requirement. If you don't have one, your combat effectiveness is non-existent.