The combination of lower birth rates, not having a socialized retirement system and not having any significant estate tax seems tailor-made to build up an aristocracy over just a few generations.
1. Because birth rates are lower, being directly supported by one's children/grandchildren is no longer feasible.
2. No socialized retirement system. As a result, each individual must accumulate a small fortune in order to retire. Most retirement plans rely on living off of the investment income without touching the principal. Therefore, the amount required to retire is also sufficient for a person to live indefinitely. (Back of the envelope calculation, assuming 3% annual return on investment after inflation, $3M in retirement savings yields about $90k/year perpetually.)
3. Currently, the first $11.18M of an estate is entirely exempt from the estate tax, and the even the top marginal rate is only 40%. At no point would it prevent inheritance of a perpetuity by several descendants.
(1) is a long-term demographic change, and I think it's good overall, in terms of the care and attention children can receive. (2) forces the accumulation of wealth, out of fear of societal abandonment in old age. (3) allows those accumulations to persist across generations. I think that the appropriate spots to break this cycle are at (2), by instituting better social safety nets and retirement care, reducing the incentive to indefinitely accumulate, and at (3) by significantly increasing the estate tax.