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

I don't see an argument being offered by your FB friend, only a conclusion. Without some reasoning, i.e., some premises, I can't tell if there are any fallacies or not. In other words, this is unsupported conclusion.

Most people that says things like this are trying to phrase an argument as a question. It is based on the slippery slope fallacy.

The reductio is used when a proposition is disproven by following its implications logically to an absurd conclusion. "Why stop at $15 an hour? Why not make the federal minimum wage $100 or $200 or $1000 an hour? What would be the downside of that?" fails to do that. In other words, suggesting an extreme on an continuum doesn't even make an ideal value on the continuum less reasonable, let alone make it absurd or "disproven." By this reasoning, we can also say that $3 per hour leads to absurdity... "Why stop at $3?..." The absurdity is in the proposed numbers in the argument, not in the inevitable outcome.

A similar example without the political bias: "Why charge .50 for lemonade at the neighborhood stand? Why not $5 or $50 or a billion dollars? What would be the downside of that?" There is no rational and calculated reductio here; it is simply a fallacious slippery slope argument where each "step" is not a logical progression from the previous one. We don't look at this "argument" and conclude "Yeah, .50 is absurd." That would be poor reasoning.

The reductio is used when a proposition is disproven by following its implications logically to an absurd conclusion

Hmmm. I'm not sure why the argument in question doesn't qualify then? Perhaps the problem is the construction we're using (from the original question). Perhaps representing the argument differently makes plain its reductio intentions?

I tried to do that with my 'MBA burger flippers,' an absurd end situation meant to follow the implications of artificially raising wages to where unemployment of the under-qualified because obvious.

Here's a different, perhaps better example from a Wall Street Journal letter to the editor:

"Please, if creating a 'significant economic stimulus' were as easy as decreeing higher wage rates, why stop at $10 or $15 per hour? Why not $20 or $50, so that everybody has lots of income?"

To your point, though, here's a different example (this one from Forbes) that isn't a reductio:

"Proponents for $10.10 [as a minimum wage] don’t exactly say where that number comes from; it’s an increase that grossly exceeds productivity strides among minimum wage workers. So tongue in cheek, I ask why not raise it to $100 while we’re in the business of throwing out random wages? Obviously no serious factions are lobbying for this, but the left is lathering up a slippery wage slope. They essentially want to impose controls on the price of labor."

Note that here, she identifies her own argument as a "slippery (wage) slope."

The question, "Why not X+Y?" is perfectly legit, assuming Y isn't absurd (or a better word, ridiculous). Consider:

I'll pay my workers $10 per hour.
Why not $15?

Now, a constructive conversation can be had as to why $10 is better than $15 or vice versa. However,

I'll pay my workers $10 per hour.
Why not a billion dollars?

Now this is just being silly and perhaps a bit childish. I would even say there is a argument of the beard fallacy here (and with the $50 -$100 retorts). Wages, like the drinking age, are set based on many factors, so it is not arbitrary or random, but it's also not objective, meaning there is some ideal value on what can be considered a bell curve.

For the argument

Please, if creating a 'significant economic stimulus' were as easy as decreeing higher wage rates...

I know this isn't your argument, but it reeks of the strawman fallacy . Perhaps someone once did claim that "creating a significant economic stimulus is as easy as decreeing higher wage rates," but I doubt it. Even if they did, there is no claim that the higher the wage, the better the results. Anyone taking this argument seriously knows that there is some ideal value for minimum wage, and as we stray from that value, positive or negative, the economic results worsen. Therefore, the entire "why not [some ridiculous amount] here?" should not even be taking seriously.

[To Bo Bennett, PhD]

I was with you until here:

Anyone taking this argument seriously knows that there is some ideal value for minimum wage, and as we stray from that value, positive or negative, the economic results worsen.

I think you are missing something important here. The premise underlying this particular argument is not there is some “ideal value” for the minimum wage. It is that a minimum wage is not ideal at all. That is, that the market should set wages, not governments. The absurdities seek to illustrate why by exaggerating the distorting effect of the government intervention.

Otherwise, I agree with you that the argument is a straw man and can certainly become childish. (ad absurdum arguments in general strike me as childish.) If making this argument, I would favor walking the other person toward my point as you suggest; i.e. by increasing the wage in more reasonable increments. If $15, why not $20? Why not $25?

I gotcha. Chalk that one up to my political ignorance. I didn't think abolishing the minimum wage all together was a position that was even seriously considered given that this has been in effect since 1938 and survived both conservative and liberal governments.

No worries. Welcome to Libertarian thinking! ;-) It's seriously considered but also realistic and patient because of exactly what you wrote: No matter who is in power, statism seems to remain the status quo.

For further reading, check out this Walter Williams article from 2007.

Or check out this more current video by John Stossel.

You are arguing the issue, not the thought process. 

We seem to be becoming from different schools of thought but reaching the same concultion; only in reverse:

https://pathosethoslogos.com/