Glen Weyl proposed a voting mechanism where every voter can vote as many times as he or she likes but is required pay each time. The trick is "the amount you have to pay is a function of the square of the number of votes you cast." Steven D. Levitt summarizes:

Just for the sake of argument, let’s say the first vote costs you \$1.  Then to vote a second time would cost \$4.  The third vote would be \$9, the fourth \$16, and so on.  A person who cast four votes would have to pay a total of \$30 (1+4+9+16=30).  Twenty votes would cost \$2,870.  One hundred votes would cost you more than \$300,000.  Five hundred votes would cost more than \$40 million. So eventually, no matter how much you like a candidate, you choose to vote a finite number of times.

Levitt defends the idea against charges that the system favors the rich:

In our existing system of campaign contributions, there can be little doubt that the rich already have far more influence than the poor.  Presumably in our current system if a rich person spent \$40 million to try to influence an election, that rich person would surely hope to change the way more than 500 people vote, whereas in this system that is all the votes you get to cast for \$40 million.  So restricting campaign spending, in conjunction with this voting scheme, might be more democratic than our current system.

Levitt acknowledges that "it is much cheaper to buy the first votes of a lot of uninterested citizens than it is to pay the price for my 100th vote," and is unlikely to be implemented any time soon. That's putting it mildly.