Is it correct to write C♯♯♯♯♯♯♯♯♯♯♯♯ (twelve sharps) as C?
It depends on the context. Even the context given in the question isn't sufficient for an answer:
the next note is always the previous note raised by a perfect fifth
How do you define perfect fifth here?
For most European music (that is, for most music using the European classical notation system regardless of the actual style of the music or its place of origin), an ascending perfect fifth must be interpreted in the context of the twelve-tone system, where it means "increase the note's letter name by four and adjust the accidental so the resulting interval comprises seven semitones." Note the absence of any explicit mention of frequency. The frequencies are determined by the temperament, but the choice of temperament does not change the definition of "perfect fifth."
On top of that, there's a somewhat flexible rule along the lines of "adjust the spelling of the pitch enharmonically for the convenience of the performer." Different composers may apply the rule differently. The same composer may apply it differently in different contexts or at different times. Jazz and popular music tend to apply it more liberally than classical music. But in every context, the rule would be applied long before you reach C triple-sharp, much less C duodecuple-sharp.
This enharmonic respelling rule is available in all twelve-tone temperaments, not only in equal temperament. In unequal temperaments, a particular tone may serve better in one enharmonic identity than in another, and nominally identical chords will therefore sound different, but the enharmonic respelling is nonetheless theoretically available. In the example song given in the question, therefore, some perfect fifths are indeed going to be spelled as diminished sixths (or perfect fourths as augmented thirds) -- if you use this definition of "perfect fifth."
The next time someone asserts that C♯ is different from D♭, you can reply "except when it isn't." For example, Beethoven's Moonlight Sonata is in C♯ minor but its middle movement is in D♭ major. Beethoven clearly did not intend for the tonal center of the middle movement to be microtonally distinct from the tonal center of the outer movements. Rather, he used D♭ major as the parallel major of C♯ minor because the notation is simpler that way. Similarly, look at the choices of keys in the 48 preludes and fugues in Bach's WTC. There are three movements in D♭ minor and one in C♯ minor.
In this context, therefore, the answer is "it's wrong to use C duodecuple-sharp, but if you really must then it's acceptable to respell it as C. Or maybe B♯. Or maybe D♭♭."
If, however, your definition of perfect fifth is "3:2 ratio of frequency no matter what" then you're not using the 12-tone system. In this case, you're going beyond what traditional staff notation can reasonably express. You're also going beyond what human musicians can reasonably play. You have a few options for notating this song, therefore.
The first is simply to describe it, as you have in the question. This would be especially useful if the song is intended as a piece of electronic music. It could then be taken, for example, as the specification for a computer program that realizes the song through audio synthesis or something like that.
Another approach would be to have several keyboards or other instruments detuned from one another by a Pythagorean comma. The first would play the first twelve notes, and the second would play the next twelve. The second and subsequent instruments would have transposed parts, so where the second plays B♯, the score shows C.
In this context, therefore, the answer is "you probably shouldn't be using staff notation for this, but if you are you should write it with a transposition to keep it legible."