It depends what your classification of "rich" is. There's a huge difference between the 10% and the 1%; an even bigger difference between the 1% and .1%.
I personally think people close to the 10% range pay their fair share in taxes and then some. They are comfortable, but not private account on retainer comfortable. They also pay a far greater percentage of their yearly income towards tax obligations than the ultra-wealthy have in decades.
The ultra-wealthy have a host of options at their disposal in order to evade paying their fair share, with the most common one being that a large chunk of their "income" is actually tied up in capital gains, which is taxed at a far lower rate than income. Additionally, the ultra-wealthy tend to "donate" a ton of money to ... unique non-profits that are tax deductible. Then there's always the offshore havens who are more than happy to hold your money with no questions asked.