In a general sense, annuities do not "run out" after ten years. There are a wide range of annuities on the market and the payout terms depend on the contract and the settlement options selected by the annuitant at maturity. As far as a ten-year period is concerned, this may be a settlement options known as the period certain.

The period certain is merely a settlement option that safeguards payments for a designated beneficiary or the annuitant's estate for a specified period. Usually, the period certain is presented as a ten year option and obligates the insurer/ annuity provider to continue to make payments in case the annuitant passes away before the ten years are up. However, if the annuitant survives the period certain, he or she continues to receive the payment for the guaranteed period of ten years. Once the ten years are up, the insurer is no longer obligated to continue payment to the designated beneficiary.

Taking a settlement option without the period certain yields a higher payout, since the insurer would not be obligated to make payments if the annuitant dies at any time. However, in selecting the period certain, the payment is reduced to cater for that possibility. For instance, a person selecting a straight life settlement option may have a monthly payment of $1,000. However, if the option selected were straight life with period certain, the lifetime payment would be $920.

Some annuities are life long annuities.
My mother had purchased one such annuity.
As long as she lives, the company has to continue the payment. I just hope that she can live beyond a hundred.