You shouldn't be getting that number with that test. Are you telling me that fPart(tanh(14.165) actually returns zero on your calculator?
When answers are given as AEB on the home screen, they are treated differently: after their orders of magnitude (B) are tested, their mantissas are then compared starting with the first nonzero digit, even if that happens to be ten or fourteen or eighty digits away from the decimal point in either direction. This was exercised by TI to handle really large or small numbers without having to operate on so many digits, and which is why 1+E-11=1 is true but 0+E-11=0 is false.
Of course, results are given approximately on the home screen, but revealed on a more system-level when shown in the STAT LIST editor. But, you also have to consider the effects of extraneous operations (e.g., subtracting from one) when the number is so small, and how that plays with the accuracy a little bit. It's better to go with the purest test first—the function and its argument, and what that looks like—with the reasoning that when this number is ultimately defined, anything else done on it will be just as though it were done on the number 1 itself.
Bottom line is, if the task is to find the smallest number that internally gets evaluated to "one" with 14 digits of accuracy, that is what is on the page right now.