Until recently, almost all bond wires were made of gold. This is because low temperature ultrasonic welding can be done reliably between gold and aluminum, and because gold and aluminum are ductile enough that they act as a cushion during the welding process to prevent oxide cracking. (In CMOS, until the 0.18u node or so---1998ish---IC interconnect was aluminum; on more modern processes, interconnect is copper, but generally for wire bonded chips there is a plating layer added to the copper bond pads to enhance bonding reliability.)
Bond wire diameters are (for historical reasons) usually specified in mils. A common bond wire diameter is 1 mil, which, depending on length, can reliably handle a bit more than an ampere without fusing. (Shorter lengths and lower temperatures help this.) Note that this number is quite conservative, since this is the fusing current at 125 degrees Celcius, and there is a quasi-exponential dependence on temperature.
(Also note that multiple bond wires can be used in parallel to increase the fusing current; in one image from the linked article [1] it appears that there are two bond wires in parallel, but kens points out below that this is a force-sense arrangement and not a double bond.)
As a result, when running a chip at room temperature, very high currents will almost always blow up the circuit rather than the bond wire. In over ten years I've only once managed to melt bond wires without also causing the chip to crater, and that was on a power converter with gigantic transistors (like, "see them with the naked eye" big) that was designed to be quadruple bonded and was only double bonded in an engineering sample.
More recently, copper bonding has become increasingly common. This is driven by higher conductivity of copper (and thus lower bond wire resistance and higher fusing current) and substantially lower cost compared to gold. However, copper bonding requires special care. For example, the bond pad structure must be built to withstand much more bonding force without cracking.
For a technology as old as the one being used to build this 7805---a mid-80s bipolar or BiCMOS fab, 1 or 2 layers of aluminum interconnect---it would probably take substantial effort to redesign the bond system for copper bonds (and there may not be enough metal layers to provide cushioning, so it might just be impossible to reliably bond with copper). Beyond that, there are only four bond wires (see the aforementioned image), so the cost is minimal (and at most bonding houses, packages come with a number of "free" bonds built into the price of the package). In addition, redesigning the chip would involve substantial engineering effort (redesign the layout, redesign the package, redo all the reliability qualifications), the cost of which would probably not be recuperated via the price difference between copper and gold.
There's an interesting reason for the parallel bond wires to the output pin, and it's not to keep it from fusing. The problem is if you put 1A through a thin bond wire, there will be some voltage drop across the wire. So if the chip produces 5V at the die, it might be 4.9V at the 7805's pin, which is no good. So they run a second bond wire from the output pin to the regulator circuit. This sense wire has hardly any current through it, so it gives an accurate value of the output voltage. Thus, the 7805 can regulate the voltage on the output pin, rather than the voltage at the die pad.
TL;DR: one bond wire to the output carries the current and the second bond wire on the output senses the voltage.
Because the heat is not on them. They're built for the nominal current of the circuit, so if you go beyond that, they'll be toast, but again, your circuit will fry first
Bond wire diameters are (for historical reasons) usually specified in mils. A common bond wire diameter is 1 mil, which, depending on length, can reliably handle a bit more than an ampere without fusing. (Shorter lengths and lower temperatures help this.) Note that this number is quite conservative, since this is the fusing current at 125 degrees Celcius, and there is a quasi-exponential dependence on temperature.
(Also note that multiple bond wires can be used in parallel to increase the fusing current; in one image from the linked article [1] it appears that there are two bond wires in parallel, but kens points out below that this is a force-sense arrangement and not a double bond.)
As a result, when running a chip at room temperature, very high currents will almost always blow up the circuit rather than the bond wire. In over ten years I've only once managed to melt bond wires without also causing the chip to crater, and that was on a power converter with gigantic transistors (like, "see them with the naked eye" big) that was designed to be quadruple bonded and was only double bonded in an engineering sample.
More recently, copper bonding has become increasingly common. This is driven by higher conductivity of copper (and thus lower bond wire resistance and higher fusing current) and substantially lower cost compared to gold. However, copper bonding requires special care. For example, the bond pad structure must be built to withstand much more bonding force without cracking.
For a technology as old as the one being used to build this 7805---a mid-80s bipolar or BiCMOS fab, 1 or 2 layers of aluminum interconnect---it would probably take substantial effort to redesign the bond system for copper bonds (and there may not be enough metal layers to provide cushioning, so it might just be impossible to reliably bond with copper). Beyond that, there are only four bond wires (see the aforementioned image), so the cost is minimal (and at most bonding houses, packages come with a number of "free" bonds built into the price of the package). In addition, redesigning the chip would involve substantial engineering effort (redesign the layout, redesign the package, redo all the reliability qualifications), the cost of which would probably not be recuperated via the price difference between copper and gold.
[1] https://plus.google.com/photos/+KenShirriff/albums/605050806...
EDIT: kens, good catch on the Kelvin connection. Now that I look at the die photo again, it's clear the left one is for sensing.