I'd open by saying that limited liability and the constitution of legal personhood was only clearly defined in all major jurisdictions during the industrial revolution. This means that the earliest "companies" weren't legal entities as we know them, but a continuation of people (often family members) performing a single commercial activity. For example, Barovier&Toso are a Venetian glass company who can trace their activity continuously all the way back to one Antonio Barovier, who was a glassblower in Murano in the mid 13th century. However, the Artisti Barovier limited liability company was only founded in 1878 (later merging with the Toso Brothers firm in 1942 to form the current company). Similarly, the Torrini jewelry shop in Florence's Piazza Duomo has been open since 1369, but I'm not even sure they're a company at all, it's "just" a shop that's been handed down the generations (as astounding as that is).

Winemaking is another industry where we can go really far back if we consider "Baron so-and-so has been making wine on this land since the twelfth century." But again, that's not a company as we know it. Although there are commercial vineyards today touting millennial heritage, for most of their history they were just where some guy grew his grapes (again, not to take anything away from the feat of a single continuous activity carrying on through the ages).

The best place to look for old "companies" (associations with some sort of personhood) is with old religious institutions and orders, especially those partaking in moneylending.

Banca Monte Paschi di Siena claims to be the oldest financial institution in the world. It was officially founded in Siena as a "Mount of Piety" (religiously affiliated microcredit institution) in 1472. However, the foundation of an agriculture-focused credit institution is cited in a statute emitted by the Siena city council in 1419, meaning that it might have existed in a different form much earlier.

There are older "Mounts of Piety" in Italy, but they're no longer in operation.

Other interesting banking groups that I know of are the Berenberg Bank, which was constituted in Hamburg in 1590 by two brothers fleeing the eighty years war in the low countries. It is also not improbable that the Berenberg family had been partaking in banking prior to this date. Today the Berenberg group is active and well in all facets of banking, but they never even implemented limited liability (all partners are still personally liable to this day).