Is it permissible to supplicate for the regrowth of amputated limbs, and is such a supplication considered an transgression or a request for something unreasonable, and are there recommended formulas for that?

The supplication for Allah to compensate you with real organs has been mentioned previously in Fatwas numbered 120202 and 116501.

It is advisable to supplicate to Allah to bless the remaining organs and to compensate for the affliction, remembering that calamities are an expiation for sins, based on the Prophet's (peace be upon him) saying: "No fatigue, nor disease, nor sorrow, nor sadness, nor hurt, nor distress befalls a Muslim, even if it were a thorn that pricks him, but Allah expiates some of his sins thereby." And his saying: "On the Day of Resurrection, the people of well-being will wish that their skins had been cut to pieces with scissors in the world, when they see the reward of the people of affliction."

A believer is patient and content with Allah’s decree, and sees mercy in affliction, based on the Prophet’s (peace be upon him) saying: "The greatest reward comes with the greatest trial. When Allah loves a people, He tests them. Whoever is content (with Allah's decree) will have contentment, and whoever is discontent will have discontent."

The glad tidings for the afflicted and the matters that help overcome calamities have been mentioned previously in Fatwas: 38905 and 32446.

Among the general supplications is the supplication of Yunus (peace be upon him): "There is no deity except You; exalted are You. Indeed, I have been of the wrongdoers," based on the Prophet's (peace be upon him) saying: "No Muslim servant supplicates with the supplication of Dhun-Nun (Yunus) concerning anything except that Allah responds to him."