Our Jeep was having problems turning over, so we called a tow truck driver. He connected the jumper cables incorrectly, starting a small fire on the battery box. After the fire was out, the Jeep no longer turned over. Besides damaging the electronics IN the engine, is it possible that he killer the starter? The battery isn't quite 2 years old, and works great. Radio, turn signals, power windows/locks & lights work fine.
More than likely the resultant issue is a fried fusible link which attaches the ignition to the starter or the starter relay which would do the same. This probably would not have killed the starter. This is the reason for the relays/fusible links in the first place. Since you didn't put down the model or year of the Jeep, it is a little hard to help you with further diagnostic as far as where to find a fusible link or as to which relay to check.
Im not an expert, so take my advice with a grain of salt, but I think I can tackle this question:
Possibly, but not likely. When you attach the wrong cables to the wrong nodes on a battery, it results in a very high current flow to the battery with a lower charge. The fuses in your fuse box are designed to burn out and break in the event of high surge of current in order to spare your vehicles components from frying. Basically how a surge protected power strip works or how the old screw-in fuses that were common in houses until breakers become standard.
Most of the time the fuses do their job correctly. However, on very rare occasions a fuse will weld itself together, creating a constant connection between battery and component.
This actually happened to my mother in law's truck a couple weeks ago. Her vehicle wouldn't turn over, however all the lights were working so the battery still had a charge. We figured the next logical thing to check was the starter. Oddly enough, when we replaced it, the truck started immediately after connecting the cables to the starter - the keys weren't even in the ignition. We swapped out the fuse for the starter in the fuse box and it turned over only when we cranked the keys in the ignition.
Check your fuses first, and if the Jeep still isn't turning over after replacing them, then start troubleshooting other damage from there.
Easiest way to tell if an electrical component is damaged is with a charge tester. Touch the positive and negative nodes to the corresponding connections in the component and it will tell you if they're still allowing current to flow through properly. You can get one at any hardware store, or swing by a mechanic or auto parts store and they will likely test out your component for free.
Or you could just see if the Jeep will crank once you've replaced the battery and repaired anything that might have burned up :)
If one connects a strong battery to a vehicle's electrical system with reversed polarity, nearly-unlimited current will flow through the alternator until it blows a fuse or fusible link, melts a wire or diode, or does something else catastrophic. Reverse-connecting a strong battery to a nearly-depleted battery that was itself connected properly would do likewise, though the depleted battery would pass much of the current itself (likely doing significant damage to itself in the process) thus reducing the amount of current flowing elsewhere. Note that reverse-connecting two batteries of equal strength cause damage equivalent to short-circuiting one battery. If the batteries are of unequal strength, the weaker battery will be damaged more severely from that experience than it would be from a short-circuit, though if the weaker battery properly tied to its vehicle's electrical system enough current might detour through the electrical system to reduce the damage sustained by the weaker battery.
If a hired "professional" connects a charging apparatus with reversed polarity, I would suggest that the customer should be entitled to have his vehicle inspected for electrical damage by a mechanic of his choosing, with the hired professional being responsible for the actual cost of his inspection or a typical mechanic's cost, whichever is less, along with the cost of repairing any electrical faults that are discovered and for which the reverse connection would be the lost likely cause. Note that such an inspection should not only try to figure out the causes behind observed problems, but even more importantly should look for damage which has no symptoms but could undermine safety. If a wire gets hot enough to destroy most of the insulation, such damage might have no immediate effect, but could lead to a later short circuit that might start a fire. If a wire's insulation is compromised in a fashion that might affect safety, the "professional" should pay to have it replaced, without regard for whether it's causing any immediate trouble.
I believe there are too many variables to say for sure. It would depend on how long everything was connected incorrectly, how much damage the fire did along with what circuits were involved. Circuits that are not normally related to the starting or charging systems may have been damaged by the fire or by feeding power thru the ground side of the circuit. I would recommend you have the electrical system evaluated by either the dealer or a shop that specializes in automotive electrical systems. You may have damaged wiring that will be a problem later. Then send the towing company the bill.
Connecting the jumper cables the wrong way might actually damage your vehicle.
If your battery is running low (that's why you can't start) then you have less voltage across it's nodes. When you connect another battery with the exchanged nodes, the current will flow out of your battery (since it's voltage is lower) and damage the battery. By the way this is a short-circuit -> The current flows unthrottled through the wires which heats everything up causing a fire. Assuming your battery is fully depleted after some time, what will be left is a battery (only one left) connected vice versa to your car. Not every component in your car can withstand this (some parts - like radios have a protection circuitery). Fuses can only limit the currrent flow, but not it's direction.