My question is about the "~たい"-form ("want to").  
I know that in a sentence containing this form, you can use は to mark the
person who desires something. For example, "私は、東京に行きたい".

But then I came across this sentence:

> あなたが一番したいことはなんですか？

I translated it as "What is the thing you want to do most?". So here, が is the
particle used to mark the one who desires something.

Now, for my question: Can が always be used to mark the one who desires
something in a "~たい"-sentence, or is it something that is limited to relative
clauses such as the one in the second example?

In extension, is it okay to say "私がケーキを食べたい" or "私がケーキが食べたい"? Sounds a bit
strange to me, but then again, my Japanese isn't really great yet.  
But anyways, thanks for reading through my question!
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particles particle-が relative-clauses i-adjectives subordinate-clauses
"~たい"-form and associated particle usage
> 「あなた **が** 一番{いちばん}したいこと **は** なんですか？」

This sentece is perfect in every way and it means what you said -- " **What is
the thing you want to do most?** ".

> Can が always be used to mark the one who desires something in a
> "~たい"-sentence, or is it something that is limited to relative clauses such
> as the one in the second example?

It is the latter.

「あなた **が** 一番したい」 is a relative clause that modifies the noun 「こと」. Inside
relative clauses, the topic/subject-marker is **_always_** 「 **が** 」. This 「が」
just cannot be replaced by a 「は」.

To use 「あなた **は** 」(instead of 「あなた **が** 」) correctly to say practically the
same thing would be to say:

「あなた **は** なに **が** 一番したいですか？」

Using 「は」 is correct and natural because it is not used inside a relative
clause or any kind of sub-clause in the sentence just above.

> is it okay to say "私 **が** ケーキ **を** 食べたい" or "私 **が** ケーキ **が** 食べたい"?

**_Only when the context calls for it, yes_**.

Let me first talk about English. Consider these two short conversations.

1) A: Who are you? B: I'm Bob.

2) A: Which one of you is Bob? B: I'm Bob.

Did you read B's two replies with the same intonation? I highly doubt it even
though I know almost no English.

In Japanese, in 1) B would say 「ボク **は** ボブです。」 and in 2), 「ボク **が** ボブです。」.
Surprised?

Back to your sentences..

If I asked you 「何が食べたいの？」(What would you like to eat?), you should reply 「ボク
**は** ケーキが食べたい。」 The focus is on what kind of food rather than who the eater
is.

If, however, I asked 「誰{だれ}がケーキを食べたいの？」(Who wants to eat a cake?), you should
reply 「ボク **が** ケーキを食べたい。」 The focus is on **_who among the group_** , not
what s/he wants to eat.

I know this is not easy, but **you will get there one day**. ← That part in
bold was a 「は」 sentence, BTW. J-learners use 「が」 **_way_** too often and I
have never quite understood why.
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You are correct about the first sentence, but I've always seen the relative
clause used. This is in a bit of a contrast to the Ⅹがほしい construction. If you
haven't learned that form yet, you might later! It's also used to describe
wants, but in a different way. So also try to keep the two forms straight!

In addition to the answer detailed about は and が , a resource that explains
the semantic distinction very well is Jay Rubin's "Making Sense of Japanese".
It's a collection of essays by Rubin, but a major point of his is explaining
the difference in context, which is often not conveyed well by Japanese
language learning classes to start with and so many of us J-learners run
across not knowing when to use which one appropriately.
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