I'm not a tax expert so I can't answer the question on why universities would do that. As far as Apple goes, I'll assume they'd like to at least beat inflation and holding cash is usually done for regulatory reasons.

Apple is different than Stanford. Stanford's fund is not an IRA - there is no tax either when they withdraw from the fund. Apple has all that money stashed because they can't repatriate it to the United States and their shareholders, without realizing corporate income tax on it in the US. So as long as they keep it overseas, the value of their stock goes up because they control the cash, but they can't award it back to their shareholders as dividends without biting the dividend tax bullet.