User blog comment:Edwin Shade/A Proof/@comment-1605058-20171104122735/@comment-30754445-20171109110527

While the example you've given is indeed infinitely differentiable, it achieves this by grinding the function's growth to a halt at every integer (all the derivatives there are zero).

I doubt many people would consider such behavior to be acceptable for a "real tetration" function. One would expect such a function to grow steadily faster and faster as b gets larger, no?