As much as you want. There are no federal or State laws that restrict the amount you can gift to children, grandchildren or anyone else. Many clients have announced to me "I can give $10,000 (or $12,000 or $14,000) to my children". Their accountant told them, or their hairstylist or a friend. In the context of federal gift tax laws, the correct statement is "I can gift $14,000 to each of my children (or grandchildren or friends) as a tax-exempt gift each year". The current federal tax-exempt gift of $14,000 can be given to an unlimited number of people, no matter how they are related to you. You may make these tax-exempt gifts every calendar year. There is no need to file a Gift Tax Return to report the gifts and there is no gift tax imposed. In addition, you can gift $5,340,000 during your lifetime without paying gift taxes. In both cases, the person receiving the gift does not have to pay any type of tax on the gift or report it on their income tax return. In another context, gifting matters very much. If you are looking at a long stay in a nursing home, the gifts you made in the last five years may disqualify you for medicaid (MassHealth) benefits to pay for your care.
If have gifted $2,000,000 to your children, you probably kept enough money to pay for many years of nursing home care. The rest of us (the 99%) do not have enough money to pay $400/day for nursing home care. At some point, your money will run out and you will have to apply for Medicaid benefits to pay for your care. Do not confuse Medicaid with Medicare. Medicare pays limited benefits for nursing home care. See Medicare, Parts A, B, C and D. The Medicaid program has financial eligibility rules and "resource transfer" rules. A resource transfer is a gift. When you apply for Medicaid benefits, you must report any gift made in the last five years. If you have, you will be disqualified from receiving Medicaid benefits. The disqualification period will depend on the size of the gift. There are exceptions to this rule and there are "hardship exemptions", but most gifts create a disqualification period. For more information about the financial eligibility and resource transfer rules, see Planning for Long Term Care 2014.
The government cannot limit the amount of gifts that you give, but some State and federal programs penalize you for making gifts by disqualifying you from receiving benefits. Before you make a gift, be sure you are aware of the consequences and your responsibility for reporting the gifts. For more information, see Estate & Gift Taxes and 2014 Changes to Income, Capital Gains, and Gift and Estate Tax Laws.