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:

[![enter image description here][1]][1]

Now that's a computer on a breadboard (well, several breadboards), built from *basic components*.  The description of it [is here][2].  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-8][3] was the size of a [small refrigerator][3]).

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][4].  Here's a [home-built calculator][5] 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.


  [1]: https://i.sstatic.net/IAHUK.jpg
  [2]: http://www.instructables.com/id/How-to-Build-an-8-Bit-Computer/
  [3]: http://www.technikum29.de/shared/photos/rechnertechnik/dec/pdp8-fluegel.jpg
  [4]: http://datamath.org/IC_List.htm
  [5]: http://www.nixiebunny.com/malmbergcalc/calc.html