How many times is it permissible to perform Tayammum (dry ablution) when there is no water or when fearing to get sick?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

It is confirmed that the Prophet said: "Soil is the ablution for a believer if he does not find water for ten years, when he finds water, he should fear Allaah, and he should pour the water on his skin.(i.e. use it for ablution)." [Al-Bazzaar]

As long as the reason for Tayammum (dry ablution) exists, which is the lack of water, or being unable to use it, it is permissible to perform Tayammum even for a long period. When one performs Tayammum, he may perform whatever obligatory or supererogatory prayers he wishes, and he is not obliged to renew the Tayammum as long as it was not invalidated, or when he becomes able to use water. 

Allaah Knows best.