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The OP wrote this comment:

I'm confused why it's "scheinen...gewesen/geworden"? I thought to form the Perfekt, you could only do that with "sein...gewesen/geworden"? Or is it because "zu sein" is included, that it satisfies that "sein" requirement for Perfekt constructions?

Answer:

The verb scheinen has also other meanings. (Die Sonne scheint = The sun is shining.) But when you use it in the meaning to seam you must use it with "zu + Infinitive":

without scheinen

1. Er schreibt ein Buch.
   ^{He writes a book.}
2. Er will ein Buch schreiben.
   ^{He wants to write a book.}
3. Er wird ein Buch schreiben.
   ^{He will write a book.}
4. Er hat ein Buch geschrieben.
   ^{He did write a book.}
5. Er hat ein Buch schreiben wollen.
   ^{He did want to write a book.}

These sentences contain a varying number of verbs. Sentence #1 contains only the main verb, #5 contains 3 verbs (hat, schreiben and wollen). These verbs belong to different classes:

• schreiben
  This is a "normal" verb which describes an action. The official name of this class is "Vollverb".
• wollen
  This is a modal verb (German term: "Modalverb"). German has 6 modal verbs: dürfen, können, mögen, müssen, sollen and wollen. There are also some other verbs that sometimes can be used as modal verbs, but that is beyond the scope of the actual topic. Modal verbs express necessities (müssen = must) or possibilities (dürfen = may) and they always modify a Vollverb, which means, that they never are the only one verb in a sentence. There must always also be a Vollverb in the sentence. (There are exceptions, where modal verbs are used alone, but this is also beyond this topic's scope.)
• werden, haben
  These are auxiliary verbs. They have no meaning on their own, they are only used to create other grammatical tenses. The verb werden is needed to create the tenses Futur I (Ich werde schlafen.) and Futur II (Ich werde geschlafen haben.). The Verb haben is needed for Perfekt (Ich habe geschlafen.), Plusquamperfekt (Ich hatte geschlafen.) and Futur II (Ich werde geschlafen haben.). There is a third auxiliary verb: sein it is used like haben, but for other verbs.

The point is, that you can have a Vollverb (which carries the main meaning), a modal verb (which turns the meaning to a necessity or possibility) and an auxiliary verb (which changes the grammatical tense) all together in one sentence, like sentence #5 from above. But only one of them can stand at position 2 which is the place for the verb that must match with the subject in person and number. It is called the "finite" verb. (I marked that finite verb bold in the examples.) All other verbs do not match with the subject (they are not marked bold), they must stand at the very end of the sentence in German sentences. (In English sentences, these other verbs stand close behind the finite verb.)

• If there is an auxiliary verb, this auxiliary verb must stand at position 2 and is the finite verb, that used the same person and number as the subject. This is the case in the sentences 3, 4 and 5.
• If there is no auxiliary verb, but a modal verb, this modal verb becomes the finite verb at position 2. This is the case in sentence #2.
• If there is neither a modal verb, nor an auxiliary verb, then there is only one verb, wich is the main verb (Vollverb). Only in this case this main verb is the finite verb at position 2. This is the case in sentence #1.

with scheinen

The tenses of scheinen and the other verbs can be different. I use scheinen always in the tense Präsens in the following examples:

1. Er scheint ein Buch zu schreiben.
   ^{He seems to write a book.}
2. Er scheint ein Buch schreiben zu wollen.
   ^{He seems to want to write a book.}
3. Er scheint ein Buch schreiben zu werden.
   ^{He seems to be going to write a book.}
4. Er scheint ein Buch geschrieben zu haben.
   ^{He seems to have written a book.}
5. Er scheint ein Buch schreiben gewollt zu haben.
   ^{He seems to have wanted to write a book.}

^{Now each sentence has one verb more than before, so that in #5 now four verbs come together, with makes this sentence extremely complicated, and therefore the construction used in #5 is never used in "real life". German native speakers use other constructions to express the same meaning. (For example: "Es scheint, als hätte er vor gehabt, ein Buch zu schreiben.") For similar reasons also the construction in #3 is rarely used. But still all 5 constructions are correct.}

What you see here, is that now the verb scheinen is always at position 2 and it matched with the subject in person and number. So, it is the finite verb in all 5 sentences. And with the verb, that was the finite verb in the versions without scheinen happened exactly the same in all 5 examples: The word zu was added, and it turned into its infinite form. This is called "zu + Infinitiv".

1. schreibt → schien ... zu schreiben (Vollverb)
2. will → schien ... zu wollen (modal verb)
3. wird → schien ... zu werden (auxiliary verb)
4. hat → schien ... zu haben (auxiliary verb)
5. same as 4

In case of "your" sentence:

ist → schien ... zu sein (auxiliary verb)

Der erste ist wirklich verrückt geworden.
Der erste scheint wirklich verrückt geworden zu sein.

Er wird verrückt.
Er scheint verrückt zu werden.

He becomes crazy.
He seems to become crazy.

Er wird verrückt.
Er scheint verrückt zu werden.

Er bleibt verrückt.
Er scheint verrückt zu bleiben.

He becomes crazy.
He seems to become crazy.

He stays crazy.
He seems to stay crazy.

The verb "scheinen" is not an auxiliary verb. Neither in German nor in English (scheinen = to seem). German has exactly 3 auxiliary verbs:

Haben, sein und werden
sind die drei Hilfszeitwörter auf Erden.

to have, to be and to become
are the three auxiliary verbs on earth.

The verb "scheinen" is not an auxiliary verb. Neither in German nor in English (scheinen = to seem). German has exactly 3 auxiliary verbs. I learned this rhyme at school:

Haben, sein und werden
sind die drei Hilfszeitwörter auf Erden.

To have, to be and to become
are the three auxiliary verbs on earth.

("Zeitwort", literal "time-word" is the German term for "verb".) Here are examples of how to use them:

• haben and sein are used to create past tenses, like Perfekt:

  Ich habe geschlafen.
  Ich bin gefahren.

• You need werden for future tenses like Futur I:

  Ich werde kochen