Bill Gates is only going to buy so many TVs, cars, and houses. Doubling his wealth is not going to change his spending habits.
It won't change his personal consumption habits, no. Instead he would spend the extra wealth on capital investments, which improve the efficiency of production and allow vast numbers of other people to buy more TVs, cars, and houses at lower prices.
Consumption is important, of course, but all economic progress comes from the part of our income which we don't immediately spend on short-lived consumer goods.
You're also conflating wealth with money. They aren't the same thing. Bill Gates has a lot of wealth, but most of it isn't in the form of money. Instead, he owns shares in various investments. Even if it were, people would simply switch to an alternate form of currency long before the symptoms you're attributing to "wealth concentration" became apparent.
There is one factor which could have the effect you're referring to, but it only occurs in non-free markets. It is not always the case that more capital investment equals lower prices; over-investment can create a situation where the prices necessary to cover the cost of the capital are above the optimal point on the supply & demand curve. Imagine, for example, that you can sell 10,000 of a certain widget per year at an optimal price point of $10 each, and that your cost per widget is $8 (for $20k profit). If you rent a machine for $50,000/year you can lower production costs (excluding the rent) to $5 each. Under normal circumstances this would be an uneconomical investment, since the rent would raise the overall cost to $10 per widget, eliminating your profit. (Raising the price is not an option since that would reduce both the quality sold and overall revenues.) However, if a regulation were introduced requiring the use of this machine it would change the parameters: now the optimal price point (for widgets produced with this machine) is perhaps 8,000 units at $18, and you only make $14,000 in profit while 2,000 potential customers do without widgets. The same argument can be applied if the machine instead comes with a tax-funded subsidy rather than a mandate. With the subsidy included, the real cost per widget is still $18, but the buyer only sees $10 at the point of sale and the remaining $8 is spread among all the taxpayers—reducing the money they have left over to spend on widgets and other goods. Any market intervention (regulation, tax, subsidy, monetary policy, etc.) which either rewards or punishes capital investment, and thus shifts the allocation of resources away from the optimal balance, reduces the efficiency of the market and the purchasing power of the average citizen.