This would be too many to list, and most resulting items would be unbalanced.
In summary, the number of possible weapon tags, damage dice, damage types and proficiency categories means that the number of unique combinations of these runs into the thousands, and most of those would be significantly better or worse than an existing weapon.
However, I give a reasonable subset of unused weapon tag combinations further below.
Additionally, many unique weapons can be made simply by taking a standard weapon and switching the weapon damage type (bludgeoning, piercing, slashing).
The existing weapons list comprises 37 items.
Each existing item has one of 8 possible damage dice (no damage, 1, d4, d6, d8, d10, d12, 2d6), three damage types (bludgeoning, piercing, slashing), two attack types (melee and ranged), two proficiency categories (simple and martial), and three mutually exclusive weight categories (light, heavy or neither).
Ignoring special weapon tags, this already gives us 288 unique weapon combinations.
Next, there are eight weapon tags, one of which is Special, which could have unlimited possibilities so I will ignore it. Of the remainder:
- Ranged weapons can have two-handed or not, and all have either ammunition or thrown, giving 576 possible ranged weapons.
- The 288 that have ammunition can either have Loading or not, giving 576 ammunition weapons.
- There are five standard ammunition-weapon ranges, giving a total of 2,880 unique possible ammunition-based ranged weapons.
- Of the 288 thrown weapons, there are three different given ranges, giving 864 thrown ranged weapons, for a total of 3,744 unique ranged weapons.
- Melee can have two-handed or not, versatile or neither; finesse or not. This gives us 576 unique combinations.
- Melee can also be thrown with three ranges, giving each melee weapon four variants: not-thrown, and three thrown. This gives us 2,304 total melee weapons.
In total, we have 6,048 weapon combinations, of which 6,011 are unused. 99.39% of the weapons you come up with are going to be unused combinations.
And nearly all them are going to be unbalanced: dealing more damage than an existing martial weapon with the same traits; having superior traits to a weapon with the same damage; being as good as a martial weapon while being simple; or being weaker than any existing weapon in its proficiency category.
What if we consider properties only?
User inthemanual suggests that the search space could be significantly reduced by ignoring damage dice and using only weapon tags.
The smallest practical set we can make involves ignoring traits which never apply to any weapon (e.g. no melee weapons with ammunition), illogical combinations (e.g. no loading thrown weapons or light+heavy), range modifiers, damage type, and the "special" property:
- Melee weapons can be finesse, light/heavy/neither, reach, thrown, and two-handed/versatile/neither, for 72 combinations
- Ranged weapons can be ammunition/thrown, light/heavy/neither, finesse, and two-handed; and ammunition weapons can be loading, for 36 combinations
This gives us 108 possible valid weapon tag combinations, not counting simple/martial. Of existing weapons:
- There are 14 unique tag combinations on melee weapons
- There are 8 unique tag comibinations on ranged weapons.
This gives 86 unique unused tag combinations, which seems reasonable. However, many of those are merely absurdly large combinations, like "finesse light reach thrown two-handed". No actual melee weapon in the book has more than three tags, and no ranged weapon more than four. Additionally, it's impossible for a weapon to be two-handed and light, since light means it can be used well in two-weapon fighting.
Taking this into account, the full list of unused melee weapons with three or fewer tags is as follows:
- Finesse, heavy
- Finesse, heavy, reach
- Finesse, heavy, thrown
- Finesse, heavy, two-handed
- Finesse, heavy, versatile
- Finesse, light, reach
- Finesse, light, versatile
- Finesse, reach, thrown
- Finesse, reach, two-handed
- Finesse, reach, versatile
- Finesse, thrown
- Finesse, thrown, two-handed
- Finesse, thrown, versatile
- Finesse, two-handed
- Finesse, versatile
- Heavy, reach, thrown
- Heavy, reach, versatile
- Heavy, thrown
- Heavy, thrown, two-handed
- Heavy, thrown, versatile
- Heavy, versatile
- Light, reach
- Light, reach, thrown
- Light, reach, versatile
- Light, thrown, versatile
- Light, versatile
- Reach, thrown
- Reach, thrown, two-handed
- Reach, thrown, versatile
- Reach, two-handed
- Reach, versatile
- Thrown, two-handed
- Thrown, versatile
However, some of these are still somewhat hard to imagine, such as finesse two-handed weapons or finesse heavy, although I suppose the spiked chain might fall into one of these categories.
Of all unused ranged weapons, excluding "finesse":
- Ammunition, heavy
- Ammunition, light
- Ammunition, loading, heavy
- Thrown, heavy
- Thrown, light
- Thrown, two-handed
- Thrown, two-handed, heavy
With only finesse weapons:
- Ammunition, finesse
- Ammunition, finesse, heavy
- Ammunition, finesse, light
- Ammunition, loading, finesse
- Ammunition, loading, finesse, heavy
- Ammunition, loading, finesse, light
- Ammunition, loading, two-handed, finesse
- Ammunition, loading, two-handed, finesse, heavy
- Ammunition, two-handed, finesse
- Ammunition, two-handed, finesse, heavy
- Thrown, finesse, heavy
- Thrown, finesse, light
- Thrown, two-handed, finesse
- Thrown, two-handed, finesse, heavy
Additionally, for any of these, or any of the game's 37 base weapons, you can change the damage type between bludgeoning, piercing and slashing, which in many cases creates a unique weapon. You also have a lot of variety in terms of the range property on thrown or ranged weapons.
Bear in mind also that you must benchmark the weapon damage based on other weapons of its proficiency class. Most reach weapons are heavy and two-handed, and those that aren't either have a major special drawback (lance) or deal tiny damage (whip).