You're absolutely correct about RF noise. In the US, the FCC limits the power of all spurious emissions to be at least 43dB below the power of the fundamental frequency. Typically transmitters use a low pass filter (e.g. Buttersworth or Chebyshev) to reduce harmonic content.
Some micro transmitters (at the very low mW level) omit the LPF entirely. Looking at the datasheet for the ATtiny85, the maximum output current is 40mA and maximum output voltage is 5.5v which means this transmitter could theoretically produce an output of 220mW. PWM will reduce this obviously. The harmonics should be below this value, but they should probably add a filter. I've transmitting across the Atlantic Ocean at those power levels using a sub-optimal antenna and WSPR (https://en.wikipedia.org/wiki/WSPR_(amateur_radio_software)) encoding.
If you cut back the antenna length so you can’t hear the fundamental 100 ft away, the harmonics shouldn’t be an issue. But, I’d have to build it to be sure. I might do that!
Here’s some info on suitable low pass filters for $5. You’d have to modify the 600 meter kit for 555 kHz, after reading up on the design at the linked technical pages.
Note that, for a given too-short antenna length, it will be better at emitting harmonics than the fundamental.
E.g. a 1m antenna is far below resonant for 1MHz AM, but may be pretty good at emitting any 37, 39, 41MHz ... content.
Of course, we expect that content to have less than 1/40th, of the fundamental, both because of Fourier series expansion of a square wave and because the frequency response of the system rolls off (edges off an ATmega aren't perfect, even before you get the reactance of the breadboard in the way.
A primary reason to add a filter, IMO, is to raise the voltage of the fundamental at the antenna, and therefore get a little more RF out on our desired frequencies.
The thing is, ... anyone who's playing with microcontrollers and longish wires is making similar RF noise.
Efficiency of, and coupling to, this antenna is going to be garbage, even for harmonics. And the breadboard itself provides a fair bit of attenuation of the already-small 10MHz-and-up content.
> the maximum output current is 40mA and maximum output voltage is 5.5v
Looking at the maximum power you're supposed to draw from a pin isn't useful. The output transistors can provide much more: it's just not good for the part.
It'd be better to consider the radiation resistance of the wire and the AC voltage driving it.
Some micro transmitters (at the very low mW level) omit the LPF entirely. Looking at the datasheet for the ATtiny85, the maximum output current is 40mA and maximum output voltage is 5.5v which means this transmitter could theoretically produce an output of 220mW. PWM will reduce this obviously. The harmonics should be below this value, but they should probably add a filter. I've transmitting across the Atlantic Ocean at those power levels using a sub-optimal antenna and WSPR (https://en.wikipedia.org/wiki/WSPR_(amateur_radio_software)) encoding.