In English,we have "I'm" instead of "I am" . Is there something like that in German?
I don't want to speak so formal!
ich bin is neither formal nor informal. It.. just is. So you can use ich bin in every situation - written or spoken. The same applies to Er ist, du bist etc.
Nevertheless, as already mentioned, German does have contractions and colloquially used short forms. To mention some of them:
Was gibt's? = Was gibt es?
Wo bist'n du? = Wo bist denn du? (equal to "Wo bist du denn?")
Gib mir 'nen Keks. = Gib mir einen Keks.
Spricht hier wer Englisch? = Spricht hier jemand Englisch?
Es ist alles nicht so einfach, weißte? = ..weißt du?
Da hab ich / Ich hab kein Problem mit. = Ich habe kein Problem damit.
Especially in written instant communication you can omit the personal pronoun is some situations:
Bin Brötchen holen.
Hast(e) mich gestern im Fernsehen gesehen?
The only reason I am can sound formal in English is that for entirely phonetic reasons the contraction I'm is used a lot. There are no such phonetic reasons for contracting ich bin, consequently there is no such contraction, and consequently ich bin never got a chance to sound formal.
Of course German also has contractions, they just affect different word combinations:
Therefore a confused German speaker trying to learn English might ask:
In German, we have "überm" instead of "über dem". Why is it that in English everybody says "over the" without any contractions? I don't want to speak so formal!
The important thing to understand is that different languages differ in where you have to choose between two connotations (e.g. choose between a more formal version and a more informal one) and where the problem doesn't even arise. That's the main difference between languages: not which details you can express if you want, but which you can leave out and when.
There is something similar. In colloquial language you can omit the "ich", so you can say
Bin gerade sehr beschäftigt.
Ich bin gerade sehr beschäftigt.
Even in colloquial conversation with German friends, you hear "ich bin" all the time.
You’d never see it in writing, but in direct speech „ich bin” will often sound rather like „ch’bin“. So you can do that, if you’re confident in your „ch”!
Additionally, I want to reinforce what has already been mentioned: Omitting the „ich“ entirely is done very frequently, even in colloquial texting (SMS to friends etc). E.g. „Bin gleich da“, which might even be phrased as „Gleich da!“, „Gehe jetzt schlafen“… In the latter case, this can also sound like an order, so mind the context! In fact, it’s not an uncommon misunderstanding (especially in text) for someone to say something like „Geh noch Brötchen holen“ and the other side assuming they’re meant to fetch some themselves.
As it hasn't been mentioned by the others. One thing I want to add:
The only contraction I can think of for "Ich bin" is "I bi(n)" as used in South German dialects (Bavaria/Austria). We use "I bi scho do" for "Ich bin schon da".
But that is only used in spoken language. I cannot think of any written example (except folk literature, but that's special anyway...)
Source: I live in Austria.
One aspect hasn't been made clear enough yet, I think:
The English contractions discussed here are pretty much standard in spoken English (less so in written English). It is considered markedly formal to use the uncontracted forms in normal conversation (exception: for emphasis, as in: "I am sure" or similar).
In German, it's the other way round: contractions are markedly informal.
So, since your question seems primarily concerned with style ("register"), I should point out that the German "ich bin" is usually on the same stylistic level as "I'm". Using contractions in German normally is on the same level as using "ain't" in English - i.e. a contraction that is markedly informal even there.
[Speculation: I can't back this up with references, but I suspect that the difference in formality/acceptance might have to do with the fact that German speakers/listeners are used to "get what they see" on the phonetic level. English speakers/listeners are trained not to rely on words looking like they sound.]
You can't expect that every contraction that is possible in English is possible in another language. There is no contraction of ich bin and it is not formal, it is the only way you can use it.