Lawrence of Puzzlania

The strategy to solving this dilemna revolves around correctly dropping supplies on your exit path to reach safety, and then doubling back over your steps.

That said, the answer to the second question, "If the desert was more than a hundred miles in radius, what's the furthest you could travel given unlimited time?" is "An infinite distance". Go twenty miles, drop one days worth of food, and go back. By repeating this you can effectively create a new oasis twenty miles away.

The first part of the question is trickier. In fact, so tricky the Minotaur missed it.

You can escape from desert in under twelve days. Here's how it might work...

On the last leg of your journey you want to leave the desert with no food. I.e., you want to spend the last three days walking out without stopping or back-tracking. To accomplish that, you'll need three days of food at a point 60 miles from the end. Let's say that's Camp Alpha.

How do you end up with three days of food at Alpha? Start one day away from Alpha (which is 80 miles from the edge of the desert) at a Camp Beta, walk to Alpha and drop off one day's worth, and return to Beta on the remaining day's supplies. On your next trip, you again arrive with two days supply. This, combined with the day's supply from the previous trip is enough to get you out of the desert. Note the trick is that you didn't have to plan for a return trip to Beta.

Now you see that at Camp Beta you'll need six days of supplies. To get that, you'll need a Camp Delta 12 miles (1/5 of a full load) from Beta. Walk from Delta to Beta and drop off 3/5 and return. Repeat and you now have 6/5 of a load. On your next trip, you once again plan on not returning and arrive with 4/5 of a load making a total of 2 loads, or six days of supplies.

So Camp Delta is 92 miles and from freedom, and we're going to need nine days (three loads) worth of supplies there. This time we're only 8 miles from the oasis so creating another camp would be inefficient. Getting from Camp Oasis to Delta and back requires 4/5 of a day of supplies. The remaining 2 1/5 days can be deposited. After two trips we have 4 2/5 days at Delta. On the third trip we have 6 3/5 days. On the fourth trip we have more than 9 days worth. Over the first three round trips you consumed 12/5 days of food. The last trip was a one-way, so that brings it up to 14/5 consumed. You now have the nine days of food at Delta required to leave the desert (you were trying to leave the desert, remember?). So in your flight you consume 11 and 3/5 days of food, so therefore that is how long it takes to escape.

All fine and good, until someone came up with a better way.