How Often Can Someone Be Anointed?

Without knowing all the particulars of your situation it is impossible to render a judgment of whether the priest’s decision was prudent.   Here is the guidelines the church issues for the sacrament of anointing of the sick:

This sacrament can be repeated if the sick person, having recovered, again becomes gravely ill or if the condition becomes more grave during the same illness. (canon 1004 §2

If a sick person who received this anointing recovers his health, he can in the case of another grave illness receive this sacrament again. If during the same illness the person’s condition becomes more serious, the sacrament may be repeated. It is fitting to receive the Anointing of the Sick just prior to a serious operation. The same holds for the elderly whose frailty becomes more pronounced. (Catechism, 1515)

Once someone begins to suffer a grave illness they can be anointed, and if their life is reasonably in danger they may receive the full Last Rites. Unless the individual recovers and then falls gravely ill again there is no need to repeat the Last Rites.