There is no "set amount of transfusions before the human body cannot produce any more blood". That statement is absolute positive nonsense.
Hemophiliacs, for example, may require monthly transfusions for their entire lives.
The "danger" of multiple blood transfusions is insignificant if you need them. With every medical procedure one must weigh risk against benefit. The word "danger" is meaningless, except as expressed within this concept.
There is a set amount of transfusions for the human body before the human body cannot produce any more blood. It may be dangerous to have multiple blood tranfusions.
The answer is no, with a caveat.
Every blood transfusion in modern times conveys a certain element of risk and complications. With the transfusion itself there is the problem of possible biological contaminants, primarily a virus or retrovirus, spoilage, and the fact there there are actually more subtypes than are categorized and there is always the risk of a slight immune reaction. Then there is the issue of repeated penetration of the vein, causing cumulative damage,which may make use of a shunt necessary. And, of course, with every entry into the blood there is the possibility (very low) of introducing infection, even with sterile technique.
The best blood transfusions, with the least risk of problems, come from a person's own blood. Some hospitals now permit one to donate and store your own blood two months before a procedure. The body will normally replenish that blood, and during an operation, if there is excessive bleeding, the physician can then use the patient's own blood to solve the problem.
Even so, blood transfusions may be necessary to save a person's life, and if there are inadequate erythrocytes and the body cannot manufacture enough to keep up with demand, then transfusions are necessary and "as many as necessary" should be administered. There is neglibible risk associated with considerable benefit.
If a physician ordered a transfusion, you should not hesitate to take the treatment.