You won't find excessive resistance in the circuit wiring by measuring continuity.

You'll want to do a Voltage Drop Test and your multimeter will be set on 12 Volts DC, not Ohms.

Also, you won't find it unless you're measuring the voltage drop under load.

So for example, If you suspected corrosion (resistance) in that battery cable, (going from the battery post to that starter relay), you'd set your multimeter to DC Volts. Put the red lead of the multimeter on the battery post, put the black lead on that cable's connection on the starter relay post, have an assistant crank the engine, and read the voltage. 

Let's say it read 3 volts. That would mean you have a 3 volt drop in that wire between its battery positive post connection  and at its connection on that starter relay. So, assuming 12 volts at the battery - 3 volts = only 9 volts going to that starter relay.

You can do the same thing to test the battery Ground cable during starting/load. In this case you'd connect one multimeter lead to the negative battery post and the other at the engine ground where that cable is bolted. Crank the engine and observe the reading.

In any case, you don't want to see much over a 0.5 volt drop.

If you did, check the connections. If they look good, replace the cable.

You can do the same thing to test the wire from the alternator to the starter relay. (Which is connected to the battery + on that starter relay lug)

Positive lead of the multimeter on the alternator lug, negative lead on its connection to the starter relay.

Engine running, turn on the headlights and the heater blower to make sure the alternator is charging the battery.  Measure the drop.

You can do the same thing on the other side of the relay that goes to the starter. (measure during cranking)

Positive multimeter lead on that lug on the starter relay, Negative on that cable's connection to the starter. Crank the car. Observe the voltage.

Don't forget the wire on that starter relay going to your power distribution box.

Here's the thing. If I followed along correctly, you're saying that the starter relay fuse keeps frying.

That's a 15 amp fuse. It only protects the wire that energizes the Starter Relay's coil when you turn the key to START. 

It's not protecting those (aforementioned) heavy gauge wires in the circuit.

That seems to point to a problem with the starter relay. Maybe replace it. Maybe that's the problem. But also, the problem could be caused by excessive heat in the relay because of voltage drop (resistance). Then replacing that starter relay would probably be a temporary fix unless you correct the underlying cause.

Also, you say you keep having to replace alternators. That points to a bad wire (corrosion under the wire insulator that you can't see). Likely either the Battery to Starter Relay wire, or the Alternator to Starter Relay wire.

Lastly, some of those heavy gauge wires have fusible links in them. I think the battery wire is protected by a replaceable main fuse (check that) but the others have fusible links built into the wires.

So to be sure none of them have fried, just check the voltages at the individual components. (Starter Relay, Starter During cranking), Alternator, Power Junction Box.) You can do this first and it isn't a Drop Down Test. Just be sure you have 12 volts at those connectors. If you do, the fusible links are fine.

So after another one of my "trademark" long-winded replies, I suggest you watch this video from Matt. He has a youtube channel called Schrodingers Box. (warning: he's even more "long-winded" than me) AND he has a White Board.  I'll post the video starting at the point when he starts explaining how voltage drop works, but if you have the time I suggest you watch the entire video.

You'll understand why testing the way you did, can fool you.

https://youtu.be/DfLyh43iihM?t=844

Well you got to have a short somewhere I would unplug various parts of the wiring harness and test them from end to end so you found where the short was and of course check all the connections for corrosion