Is it fallacious to argue that if you want a minimum wage at $X / hr, why not just make it $Z / hr where Z > X ad nauseum

Straw man and red herring: Minimum wage is not a living wage. 

Also, our economic system has no problem with profit inflation when it comes to shareholders demanding big bucks from the CEO but when it comes to wages everybody has a cow. Read up on wages during the Black Death. Sorry. Read it in a book somewhere that escapes me. But it's out there somewhere.

There is no argument here, simply conclusions.

I think your friend is starting too high in the hierarchical structure of reasoning, and you bit the bait. Before one begins discussing something (X), the parties must agree to operative definitions and agree upon the premises.

In this case, the discussion rests upon the figure of the minimum wage. Before that, you need to examine whether any minimum wage at any level is proper. To do that, you need to explore the minimum wage's purpose and see if any minimum accomplishes that purpose without creating unintended consequences that are much worse than the proposed benefit of any minimum wage.  

Dr. Walter Williams wrote several short excellent articles on the subject. You can find extensive research by Dr. Thomas Sowell, Dr. Daniel Mitchell,  or Dr. Reisman. 

This is more of a reductio ad absurdum than a slippery slope argument.

The slippery slope would argue we can't increase the minimum wage because then politicians would keep increasing it every time they wanted to score easy points with voters. That argument would sound more like: "If we go to $15 an hour, it won't stop there. Every time these politicians are up for election, they'll raise it again. Soon the minimum wage will be $20, then $30 and so on."

On the other hand, those who argue "why stop?" are trying to illustrate the fallacy of assuming that mandating a minimum wage actually helps the people it claims to help. The idea is that when you get to a high enough number, you'll realize that only MBAs would be qualified to flip burgers or some other absurd situation, and that will lead you to realize the minimum wage is actually zero -- that is, unemployment for the under-qualified.

If we wanted to assign a logical fallacy to this argument, I agree that it would be the strawman fallacy. In essence, this reductio increases the "straw" until the opposing argument is easy to knock down. Indeed, it increases the straw until the argument falls apart on its own.

We could also say this reductio is indirectly begging the question, which is: Does a $15 minimum wage do more harm or good?

Yes, you nailed it. This is a slippery slope fallacy. While the usage may be a bit different from the textbook examples, the reasoning follows the same fallacious pattern.