While waiting for the boards, I need to write the program that would run on those boards. Quick googling allowed me to find an Arduino sketch that plays the Nyan Cat tune on Arduino. A quick refactoring and fixing of formatting, and here it is:

/*
 * Nyan Cat
 * Plays Nyan Cat
 * Example: 
 * Circuit: 8-ohm speaker on digital pin 8
 * By: Bruce Helsen (geobruce), Stan Draulans (purewantfun)
 */

#include "pitches.h"


const unsigned int INTRO_NOTES[] = {
    NOTE_DS5, NOTE_E5, NOTE_FS5, NOTE_B5, NOTE_DS5, NOTE_E5, NOTE_FS5, NOTE_B5,
    NOTE_CS6, NOTE_DS6, NOTE_CS6, NOTE_AS5, NOTE_B5, NOTE_FS5, NOTE_DS5,
    NOTE_E5, NOTE_FS5, NOTE_B5, NOTE_CS6, NOTE_AS5, NOTE_B5, NOTE_CS6,
    NOTE_E6, NOTE_DS6, NOTE_E6, NOTE_B5,
};

const unsigned char INTRO_DURATIONS[] = {
    16, 16, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 16, 16, 8, 8, 16, 16,
    16, 16, 16, 16, 16, 16,
};

const unsigned int MELODY_NOTES[] = {
    NOTE_FS5, NOTE_GS5, NOTE_DS5, NOTE_DS5, REST, NOTE_B4, NOTE_D5, NOTE_CS5,
    NOTE_B4, REST, NOTE_B4, NOTE_CS5, NOTE_D5, NOTE_D5, NOTE_CS5, NOTE_B4,
    NOTE_CS5, NOTE_DS5, NOTE_FS5, NOTE_GS5, NOTE_DS5, NOTE_FS5, NOTE_CS5,
    NOTE_DS5, NOTE_B4, NOTE_CS5, NOTE_B4, NOTE_DS5, NOTE_FS5, NOTE_GS5,
    NOTE_DS5, NOTE_FS5, NOTE_CS5, NOTE_DS5, NOTE_B4, NOTE_D5, NOTE_DS5,
    NOTE_D5, NOTE_CS5, NOTE_B4, NOTE_CS5, NOTE_D5, NOTE_B4, NOTE_CS5,
    NOTE_DS5, NOTE_FS5, NOTE_CS5, NOTE_DS5, NOTE_CS5, NOTE_B4, NOTE_CS5,
    NOTE_B4, NOTE_CS5, NOTE_FS5, NOTE_GS5, NOTE_DS5, NOTE_DS5, REST, NOTE_B4,
    NOTE_D5, NOTE_CS5, NOTE_B4, REST, NOTE_B4, NOTE_CS5, NOTE_D5, NOTE_D5,
    NOTE_CS5, NOTE_B4, NOTE_CS5, NOTE_DS5, NOTE_FS5, NOTE_GS5, NOTE_DS5,
    NOTE_FS5, NOTE_CS5, NOTE_DS5, NOTE_B4, NOTE_CS5, NOTE_B4, NOTE_DS5,
    NOTE_FS5, NOTE_GS5, NOTE_DS5, NOTE_FS5, NOTE_CS5, NOTE_DS5, NOTE_B4,
    NOTE_D5, NOTE_DS5, NOTE_D5, NOTE_CS5, NOTE_B4, NOTE_CS5, NOTE_D5, NOTE_B4,
    NOTE_CS5, NOTE_DS5, NOTE_FS5, NOTE_CS5, NOTE_DS5, NOTE_CS5, NOTE_B4,
    NOTE_CS5, NOTE_B4, NOTE_CS5, NOTE_B4, NOTE_FS4, NOTE_GS4, NOTE_B4,
    NOTE_FS4, NOTE_GS4, NOTE_B4, NOTE_CS5, NOTE_DS5, NOTE_B4, NOTE_E5,
    NOTE_DS5, NOTE_E5, NOTE_FS5, NOTE_B4, NOTE_B4, NOTE_FS4, NOTE_GS4,
    NOTE_B4, NOTE_FS4, NOTE_E5, NOTE_DS5, NOTE_CS5, NOTE_B4, NOTE_FS4,
    NOTE_DS4, NOTE_E4, NOTE_FS4, NOTE_B4, NOTE_FS4, NOTE_GS4, NOTE_B4,
    NOTE_FS4, NOTE_GS4, NOTE_B4, NOTE_B4, NOTE_CS5, NOTE_DS5, NOTE_B4,
    NOTE_FS4, NOTE_GS4, NOTE_FS4, NOTE_B4, NOTE_B4, NOTE_AS4, NOTE_B4,
    NOTE_FS4, NOTE_GS4, NOTE_E4, NOTE_E5, NOTE_DS5, NOTE_E5, NOTE_FS5,
    NOTE_B4, NOTE_AS4, NOTE_B4, NOTE_FS4, NOTE_GS4, NOTE_B4, NOTE_FS4,
    NOTE_GS4, NOTE_B4, NOTE_CS5, NOTE_DS5, NOTE_B4, NOTE_E5, NOTE_DS5,
    NOTE_E5, NOTE_FS5, NOTE_B4, NOTE_B4, NOTE_FS4, NOTE_GS4, NOTE_B4,
    NOTE_FS4, NOTE_E5, NOTE_DS5, NOTE_CS5, NOTE_B4, NOTE_FS4, NOTE_DS4,
    NOTE_E4, NOTE_FS4, NOTE_B4, NOTE_FS4, NOTE_GS4, NOTE_B4, NOTE_FS4,
    NOTE_GS4, NOTE_B4, NOTE_B4, NOTE_CS5, NOTE_DS5, NOTE_B4, NOTE_FS4,
    NOTE_GS4, NOTE_FS4, NOTE_B4, NOTE_B4, NOTE_AS4, NOTE_B4, NOTE_FS4,
    NOTE_GS4, NOTE_B4, NOTE_E5, NOTE_DS5, NOTE_E5, NOTE_FS5, NOTE_B4, NOTE_CS5,
};

const unsigned char MELODY_DURATIONS[] = {
    8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16,
    16, 16, 16, 16, 16, 16, 16, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
    16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 8, 16, 16, 16, 16,
    16, 16, 16, 16, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16,
    16, 16, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 8, 16, 16,
    16, 16, 16, 16, 16, 16, 8, 8, 8, 8, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16,
    16, 16, 16, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 8, 16,
    16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 8, 16, 16, 16, 16, 16, 16,
    16, 16, 16, 16, 8, 8, 8, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16,
    16, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 8, 16, 16, 8,
    16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 8, 16, 16, 16, 16, 16, 16, 16, 16,
    16, 16, 8, 8,
};

const int BEEP_PIN = 8;
const int SPEED = 1500;


void play(const unsigned int *notes,
          const unsigned char *durations,
          const unsigned int length) {
    for (unsigned int n = 0; n < length; n++) {
        int d = SPEED / durations[n];
        tone(BEEP_PIN, notes[n], d);
        delay(1.3 * d);
        noTone(BEEP_PIN);
    }
}


void setup() {
    play(INTRO_NOTES, INTRO_DURATIONS, 25);
}


void loop() {
    play(MELODY_NOTES, MELODY_DURATIONS, 216);
}

I'm not sure it will fit onto the ATtiny85, but if it doesn't, I see a lot of room for optimization there.

Since this is just "for fun" project, I'm completely ignoring the copyrights here (the Nyan Cat is a registered trademark anyways). I'm not even sure it would apply, since if I decided to transcribe that song myself, I would come up with exactly same notes, so there is not much originality here. I'm retaining the names of the original authors, though, because I think they deserve it anyways.

P.S. Wow, I didn't know you could embed videos in code!