1. The scientific method cannot establish absolute certainty.
I have pointed this out before. As an inductive method of arriving at conclusions, the scientific method cannot establish absolute certainty in anything that it touches in the same way that a logical deduction or mathematical proof can. Inductive methodologies can only provide a probability, but not a certainty, about truths. In order to establish a certainty, scientific discoveries and principles must be paired with a deductive argument (philosophy) or mathematical proof (mathematics). It cannot do this on its own.
2. The scientific method cannot prove any unfalsifiable claim.
The scientific method cannot prove any claim that cannot be falsified. If a theory is to be considered scientific, it must be provable or falsifiable. It must expose itself to the possibility of being demonstrably false. If it does not, it is not a scientific theory, since it cannot be tested by the scientific method. The scientific method, for example, cannot prove that you are not a brain in a vat of chemicals being stimulated into thinking that you are reading this blog post right now. For all you can prove with the scientific method, the external world may not even exist at all! This cannot be verified or falsified by the scientific method, but rather belongs to the realm of philosophy.
3. The scientific method cannot prove claims about unobservable phenomenon.
The scientific method is, by definition, a method that relies on observation. If something cannot be observed, it cannot be tested by the scientific method. For example, historical events cannot be observed, and therefore cannot be evaluated by the scientific method. This does not mean that truth about such matters cannot be known. It merely means that no one who is living today was around to observe and repeatedly test, say, the presidency of Abraham Lincoln. No scientist was able to place Lincoln in the presidency, observe the results, and then remove him from the presidency to observe the results of removing him from office. Any questions surrounding the presidency of Lincoln, or of any other historical event, rightly belong to the realm of history, not the hard sciences.
4. The scientific method cannot prove certain types of truth.
I have made this point before. There are certain types of truth that cannot be accessed by the scientific method. I have already pointed out historical truths as one such truth in this post. Other truths not accessible by the scientific method include epistemological truths and legal truths. These belong to other branches of study.
5. The scientific method cannot prove its own assumptions.
If anyone attempted to use the scientific method to prove any of its own assumptions, they would end up arguing in a circle. For example, you cannot use the scientific method to prove epistemological truths, or truths about how truth is known. If you assume that truth can be known by the scientific method, you cannot then use the scientific method to prove that truth can be known. You know? Others have pointed out that logical and mathematical truths fall into this category of assumptions that cannot be proven by the scientific method. The same can be said of the reality of the external world. If you assume the reality of the external world in order to utilize the scientific method, you cannot then use the scientific method to prove the existence of the external world. I think you get the point.
In short, science is a useful tool in uncovering truth. However, it cannot be said to be the "be all, end all" method of discovering truth. It has its limitations, just as every other area of study has its limitations.