I'm going to disagree that placing a microcontroller on a breadboard qualifies as building a computer on a breadboard. Except for I/O (such as a keyboard and display), a microcontroller by itself is pretty much a complete computer. Just placing it on a breadboard and connecting up a few wires is trivial and can be done in ten minutes.
When the OP asked, "Is it possible to make a simple computer entirely with breadboards and basic electronic components?", by basic electronic components I think it means something more like this:
Now that's a computer on a breadboard (well, several breadboards), built from basic components. The description of it is here. It's made up of a dozen types of 74LS00 series IC's. (I don''t think we want to go all the way back to transistors; the original PDP-8PDP-8 was the size of a small refrigeratorsmall refrigerator).
As far as a scientific calculator goes, if you built a general-purpose computer like the one shown above, then it could be programmed as a scientific calculator. Constructing a scientific calculator using only logic IC's (no computer) would be extremely difficult; all the manufactures of calculators like that (Ti, HP etc.) used special large scale IC's. Here's a home-built calculator that uses am early 4-bit calculator IC.
I will agree that if one wants to get a computer up and running as quickly as possible, then using a microcontroller is the way to go. If one wants to really understand how a computer works internally, then building one out of basic ICs is the right path.