One of the worst is inherited fortunes. I think the world would be a much better place if extremely rich people couldn't leave a fortune to their kids. Those kinds of kids practically always end up being fail-sons or fail-daughters. Besides, if you remove the motivation of "leaving behind an empire for my kids" you probably remove the excuse for acquiring more wealth than you could possibly spend in a lifetime in the first place.
Also, this would really help with problem of "elites". If your family name didn't imply a lot of wealth, would colleges etc. be so eager to let in rich scions? IMO the number one problem america has is we have a ruling class of "elites" that know practically nothing, but think they're experts because they have degrees from nice schools and their families have money. Almost every disaster this country has had in the last 20 years have been from these various "elites", such as the economic collapse in 2008, the continued ridiculous wars since 9/11, etc. If we broke up these incestuous "elite" circles by not allowing this generational buildup of influence and wealth a lot of the inequality problems would go away also.