To answer that I would suggest an area leading to this situation is the lack of support in our education system. More people than ever before are entering college and a degree has never been as important in the workforce than it is today. Even so, education has been the scapegoat target of governmental spending effectively leading to regressive taxation through the elimination of benefits.

Let me explain, the typcial college student is entering a stage in their life when they become self dependant and financially self-reliant. With the reduction of education spending that cost now exists where it previously did not. While one can simply choose not to attend college, the inherent need is built in. Some students have the benefit of parental support to cover tuition but those costs are mearly being transferred over to the middle income families, as the lower income students despite receiving more funding through grants retain the costs in the form of debt, creating a economic group with negative net worth. This creates problems with the loaning system and loans are not within reach to many that are educated and willing to start businesses. 

Essentially maintaining a tax system that shifts the rich gets richer effect down the economic scale encourages growth, while keeping the bar high leads to stagnation and debt. Providing cheap, affordable, and perhaps free eduction allows for more opportunity for individuals. It is far more important that we fund that with our tax dollars than it is to have them be used as a welfare system that simply lets those we've left behind through mismanagement survive. We can't reduce wealfare if we are keeping everyone in the poor house.

Yes, but I am not refering to direct redistribution. The issue is the restriction on opportunity throughout the middle to lower income scales. People tend to become rich by those means but we need to ensure that individuals can access those means. The rich, at some point, were not rich and the key to economic growth is to increase that, not neccessarily or at least in part to increase the wealth of those who already did so.

I think it is appropriate you began that with "I'm" because it enables it possibly be filed under what it belongs: "Irony".

As for the picture, that has been hovering around economic thought, from since the industrial revolution, and probably after concerns of Malthus  were found to not be accurate.  That situation has been ducked for awhile now, but there is no guarantee it can't happen or won't happen.  The End of Work goes into saying it is going to happen eventually also, and we need to deal with it, through the use of NGO sector employing people to help better the community.

Of course, but how do you ensure they have the opportunity to advance if they work hard enough and invest properly?

Your answer may be quite different than mine..

No, not at all. Someone else's wealth is none of anyone's business. Just worry about yourself, not how much money Warren Buffet makes and how much tax he pays.

A recipe for ruin is everyone minding their own business and not thinking at all what is happening systemically as a byproduct of people acting as self-focused autonomous units.    Ignore how those with money will rig the system so they can take extra risks, and then get bailed out later.  The reality is that no one lives alone, and the wealth of others is dependent upon the wealth of others in an economy.  If everything goes south, then the rich person won't have much in the way of real wealth.  All that land and capital owned for production don't do much, when no one can afford to buy what they are offering.

Or, do you not see this?

I guess you could ask an alternative question ...

Suppose you live in a world with only 2 participants, a rich person and a poor person, which of the following situations is more ideal (assuming constant buying power for the dollar in both worlds):

a) Poor person earns $10,000/year and the rich person earns $50,000 per year
b) Poor person earns $20,000/year and the rich person earns $150,000 per year

Now how on Earth could there possibly be a constant buying power for both these worlds?

If in the second world the relative productivity is (roughly) 3 times the productivity of the first world

Perhaps it's merely the medication taking effect here, but something about this strikes me as being off.