Raising the minimum wage primarily has the effect of increasing the wages of the lowest earners (duh, right?). Which also means that this weakens the "strength" of the middle class relative to the poor as well. If getting a good job and working hard for a decade or so gets you to a place economically where you're making 10 times minimum wage, and they double the minimum wage, what has that done to your relative economic progress?
Where are you getting this? What magical fairy-tale land do you get a job starting at minimum wage evolves into a position making 10 times more after only a "decade or so?"
First off, I didn't say you start at minimum wage, but that's not the point. The point is that you spend time increasing your earnings. The return on that time spent is how much more you are earning relatively speaking than someone else who's just starting out. Minimum wage is a good baseline comparison to use for this.
And for the record, it does not require a fairy tale to occur. In 1996 I was earning $6.50/hour (which was just barely over minimum wage at the time). I was hired at my current job and started as a temp at $9/hour. A year later I was rolled over to regular employee and my salary increased to $12/hour. A year after that I was earning roughly $50k/year. By the ten year mark, I was earning north of $80k/year (which is ten times the current minimum in California). Today, I earn somewhere around $120k/year, not counting bonuses and stock earnings.
So yeah. This sort of thing does happen. My case may be a bit extreme, but even more moderate cases occur all the time. Again though, the point isn't about any specific ratio, but the fact that the ratio of your earnings today versus what a new worker just entering the market will earn at a minimum (aka "minimum wage") is a measurement of your own personal economic advancement. When that ratio changes in a negative direction, it does have a negative effect on you.
Not only that, but how many "middle class" families living comfortably have to hold back on spending because their "minimum wage" relatives can't afford to live on their own. Do you think it will matter if people making TEN TIMES minimum wage are butthurt over being only five times better than human beings who work for the least amount of money legally allowed?
You kinda made my point for me though. No matter how high you raise it, by definition minimum wage will always be " the least amount of money legally allowed". You aren't changing that by raising it. What you are doing is hurting the advancement of everyone else. You're also still obsessing over one case. How about a different one? Let's imagine someone earning $10/hour in a state where the minimum wage is $8/hour. He's spent some time and sweat getting to the point where he's making $2 more per hour than the guy who started yesterday, right?
What effect does raising the minimum wage to $10/hour have on him? Pretty demoralizing, right? I mean, why bother making sure to come to work on time, and work hard, and whatnot if it isn't going to make a difference in the long run?
I mentioned middle class wages because I was responding to the somewhat absurd idea that raising the minimum wage somehow "strengthens the middle class". It obviously doesn't, and if anything weakens it (slightly). But the people who really get screwed over are the working class folks. They're often making more than minimum wage, but not much more. Point being that the only people minimum wage helps are people who can't earn more than minimum wage. It hurts every single other person in the economy to some degree.