I think you can ask the question as "What is the maximum that I can give to family members per year so that they won't have to pay additional taxes?"
For US Income Tax purposes, any gift received is not income and no income tax is ever owed on the gift. Therefore, your family members will not have to pay any tax on any amount you give to them.
However, YOU will have to pay US Gift Tax if the gifts are too large. [Since you presumably would be giving more to your family if you did not have to pay the gift tax, it is just as bad for them.]
There are several ways that people escape the gift tax.
First, there is an annual exclusion of (currently) $11,000 per recipient for outright gifts . This means that the first $11,000 that you give to EACH family member is excluded from the gift tax. So, if you have a lot of family members, you can transfer a lot more money every year. [Two notes:  the $11,000 annual exclusion is adjusted for inflation by increments of $1,000 &  spouses can elect to 'split-gifts' and by doing so can each give up to $11,000 to each family member.]
Second, there is a lifetime exemption amount of $1,000,000. Although the exemption amounts for the US Estate Tax are set to go up between now and 2009, the gift tax exemption will stay at $1,000,000. This means that over the course of your lifetime, you can give $1,000,000 of taxable gifts without ever paying the US Gift Tax. Taxable gifts are those gifts that are above the annual exclusion amount. So, no part of the first $11,000 given to each individual is a taxable gift, only any amount above the $11,000. E.g. if you give $15,000 to your grandchild in a year, the first $11,000 is excluded & the last $4,000 is a taxable gift, which exempted under your lifetime exemption (unless of course you have already used it up). The bad news is that any of your lifetime gift tax exemption amount that you use up counts against your estate tax exemption. This may not matter if your estate will be sufficiently small (or if the estate tax is permanently repealed).
There are other ways to play games with the gift tax system other than simply giving under the annual exclusion & lifetime exemption amounts. If you have that much money, hire a good tax attorney to draw up a lifetime transfer and estate plan for you. Most of the plans revolve around special instruments, such as trusts, or entities, such as partnerships, to bend the tax rules a little.