There are two reasons why the Romans had good quality equipment:

1. They were willing to copy equipment used by their enemies/neighbours if it was better. For example, the late Republic saw the Romans using Celtic-style body armour (mail), Celtic-style helmets, and Spanish-style swords (gladius hispaniensis).

2. They had a relatively large metals industry, part of which appears to have been dedicated to supporting the army.

Together, this gave a combination of quality and quantity. In terms of quality alone, Roman equipment was unremarkable: armour consisted of helmet, body armour (mail, scale, or lorica segmentata), and a shield (if you include that as "armour"), and many others also used helmet, body armour such as mail, and shields.

Once the classic Roman legionary equipment was in place, it wasn't long before it was outdone by Rome's enemies, at least in quality: the fully-armoured cavalryman, with head-to-toe armour and an armoured horse - the cataphract - was adopted by the Parthians. (The Roman army then proceeded to catch up in quality by fielding their own cataphracts.)

In terms of quantity, Han Dynasty China might have beaten the Romans - according to inventories, they had sufficient armour on hand to equip the entire Roman army and have leftovers. Qualitatively, Chinese armour of the time wasn't lacking compared to Roman.

A few points are useful to note:

1. Armour much more protective than that of Roman infantry was usually cavalry armour. Weight matters! One can consider when common infantry armour became significantly more protective than Roman armour, and also consider when cavalry armour became better than cataphract armour.

2. Once armour is sufficient to keep out arrows, spears, and swords, it's doing its job, and doesn't need to be more protective. Since the power of bows was limited by the human archer, the main threats to arrow-proof armour were lances, heavy javelins, and powerful crossbows, until guns came into use.

3. The crossbow may well have been an important driver of the improvement of armour in late Medieval Europe, when crossbow-proof armour that would also stop early guns appeared. If hardened steel plate armour was used, this would have been lighter than equally-protective sandwiches of textile armour, mail, and more textile armour or combinations such as lamellar over mail. Weight matters, so being lighter than previous equally protective armours was important, and can be considered qualitative superiority. More protection at the cost of increased weight is not necessarily better (it can be, if that protection is needed, and the extra weight isn't too great).

So, when was legionary exceeded in quality by common infantry armour? If we accept better protection as sufficient for "better", even if this comes at the cost of extra weight, then 12th century mail shirts with long sleeves and mittens are better than legionary armour (hand protection is nice). Possibly earlier long mail shirts would count. If we want better protection for a given weight, then hardened steel armour definitely meets the criterion. This could include late Medieval European plate, Japanese lamellar with hardened steel-faced plates, and probably some Ming Chinese armours (which could be bullet-proof).