Usamah ibn Munqidh in the 12th century records owning a kazaghand that incorporated a long Frankish mail shirt on top of a short one. Link

There were some kinds of mail armour which seem to have had areas where the mail was 'doubled up'. The most common example is probably the shoulders of some roman mail, where two flaps run from the back to the chest. This was apparently to protect the collar-bone from being broken by heavy downward blows, although why they chose to use more mail when it seems as though a more rigid form of armour or just extra padding would serve that purpose better is a mystery to me.

Another example, from the Medieval period, is a type of Russian hauberk which opened like a coat rather than being pulled over the head in the usual manner. These were like a 'double breasted' jacket in that there was a considerable area of overlap between the two halves, so the front of the body was covered by two layers of mail.

Separate neck guards (aventails and standards) would usually overlap with a hauberk, sometimes covering most of the chest and shoulders with two layers of mail. Hauberks which had a ventail (a kind of bib hanging below the neck which could be pulled up to cover the face) sometimes seem to have been worn with them down, which would give another layer over the chest (although it seems likely this was done to make breathing and talking easier rather than to protect the chest).

While there are some cryptic references to "double mail" and "jousting mail" in historical sources, particularly Joinville, it's really hard to tell what they meant by it, as none of these have survived. They seem to have been particularly protective garments - see, for example, Joinville's servants covering him with his jousting mail while he lay sick on the deck of a ship under attack - but exactly what they constituted is unknowable. The bog standard mail coat was composed of iron rings woven 4-in-1 (that is, four rings attached to each ring). Would these heavier mail coats have consisted of smaller, more tightly woven rings, a double layer, or what? Your guess is as good as mine.

However, wearing two mail shirts is unlikely because of the weight. A full-length hauberk, of which very, very few survive, is probably in the region of 25-35 pounds. That's not counting the helmet, the mail chausses, weapons, and everything else. We know that knights and other warriors avoided wearing their armor up until almost the last minute because of the weight and discomfort. It's likely that two hauberks would have proven too cumbersome for effective use.