I seem to recall building small AM transmitters in elementary school. I don't remember if I used a transistor or not. I think it was easy to construct the capacitor and the inductor. (I build it in the early 70's and don't really remember how.)
We also made crystal diode radios, but I have no idea how to construct a diode from scratch, so maybe those are not simple.
Traditionally, one would have used galena (natural lead II sulfide) crystals. I also recall most "boys'" radios - the type you'd find in Cubs/Scouts/etc. handbooks - using a razor blade as the contact whisker, probably because it was the simplest way to get a small contact point and stiffness at the same time with materials a kid would likely have to hand.
Back when radioshack existed they used to sell kits where you could assemble a crystal radio out of a handful of simple parts. This was the first real project my dad and I ever did and I was just a little kid at the time.
One of those kits ended up being a growing up story of mine, and a lesson in unchecked hubris.
I was maybe 8-ish years old, and trying to follow the instructions to wire together a crystal radio with one of those Radio Shack electronics kits. My stepdad - who had only recently become such, and to my assumption at the time was just some dumb hick mechanic - volunteered to help me troubleshoot.
Being the bratty child I was, and being frustrated by my inability to figure it out even with the "easy" diagram (let alone actual schematics, which might as well have been hieroglyphs to me), I snapped back at him "I'm ten times smarter than you!" and stormed off.
Not 5 minutes later he calls me back in, triumphantly presenting a working crystal radio.
Turns out my "dumb hick mechanic" of a stepfather was actually once upon a time a technician for Nakamichi, and specialized in fixing their receivers. That little crystal radio was trivial compared to the sorts of circuits he used to repair for his day job.
That was 20 years ago and he still gives me shit for it, lol
The tables did turn on that more recently, when he was having trouble with his computer (an ancient-by-today's-standards Dell Optiplex running Windows XP); his power supply died, so I bought him a new one, but after he installed it he wasn't able to boot into Windows anymore. I figured maybe whatever blew the PSU also blew the hard drive (I've seen that happen before), so next time I was at his place he handed me the tower to take home and diagnose/rebuild.
As he's handing it to me, it dawns on me, "Well wait just a second..."
On the spot, I popped open the case, took a quick look at the hard drive, plugged in the Molex cable, put the cover back on, and handed it right back to him.
Hearing the words "Alright fucker, we're even" made my week.
Wow, that brings back some memories. I built one of those too. It was a crystal radio with no need for batteries. I used that little radio for years to listen in bed until I fell asleep.
You don't need to know anything about Maxwell's equations to build a functioning radio. There are plenty of do it yourself books from the 1920s and 30s that explain in great detail how to do it including how to make rectifiers for battery charging, winding coils to get the right inductance, calculating the length of wire needed for the aerial and so on, together with the simple formulae that are needed.
There's reasonably good evidence to suggest that the first radio was invented before there was widespread acceptance of these so-called "waves" that Maxwell's ridiculous maths seemed to predict - before either Hertz or Lodge had managed to prove that they were anything other than artifacts of Maxwell's electromagnetic model. A certain David Hughes demonstrated transmission over a distance, but it was just written off as induction. And one could even presume that radio would have been inevitable, even without a preexisting theory of it, based entirely on barometers lighting up in thunderstorms; it only needed Volta's pile or a simple generator - some source of current on demand - and some experimentation with electricity before the phenomenon was noted and then deliberately employed.
You can even create an electric arc between the ground and the transmitting tower as a cheap (and dangerous) receiver: https://youtu.be/uo9nGzIzSPw