So... I figured out how to, like the title suggests, RUN BASIC PROGRAMS FROM AXE!
Without more of an intro, here is the code.

.NAME
GetCalc("prgmBASIC",Y1 //gets the pointer to the basic program
[0523]->Str1           //stores the memory location of the home screen
GetCalc(Str1,{Y1-2}r) -> A
Copy(Y1,A,{Y1-2}r      //copy the program to the home screen
Asm(21F8893E02AE77)    //enables loops, if blocks, ect... on the home screen
Str1:Asm(E7EF9B4A)     //runs the program
Asm(21F8893E02AE77)    //returns to normal home screen

And that is all! Once this is compiled and run, it will execute prgmBASIC!
(Your previous entry will be the program that you just run...)
I imagine something bad will happen if the basic program throws and error, I probably should have included some kind of error handler (like the Runprgm Axiom)
I got the home screen swap hex code from here: http://tibasicdev.wikidot.com/hexcodes
Thanks Runner for the program execution hex code!

This works on my ti-84, is there any improvements I should make?

If you try to run a non existent program the screen just flashes for a second and then clears and displays done on the top right corner.
Nothing bad seems to happen. If the Basic program throws an error (Like if the user hit the ON key) the program quits the same way as usual. Overall it acts like you ran the program normally from the home screen. (with the exception of a nonexistent program)
(Learning Basic is the best place to start - that is where I began!)

So, because you are running basic thru Axe, is it as fast as Axe, or as slow as basic? For example, would DJ Omnimaga's original Reuben Quest 2 run as fast as Sorunome's Axe app? Or would there be no change in running speed?

There's no change in speed, since TI's BASIC interpreter is still actually running it. The Axe program just tells it what to run.

First, I want to say I am impressed at how you pieced that together (reminds me of my early days of hex coding).

This may be more efficient and smaller.

It's pronounced like 'Ask'   

I am working on a hybrid asm-basic program with this discovery and I am wondering, how do you disable the APD in basic while waiting for input?

Thanks!
I haven't tried the code but it has an odd number of characters.
Don't all hex codes need to have an even number?