Say you've got four 4s -- 4, 4, 4, 4 -- and you're allowed to place any normal math symbols around them. How many different numbers can you make? It's best to think of a number and then try to make it.

The origination of the puzzle and some examples are in this video

I'm not quite certain if this is allowed by the problem such as using variable x, such as using standard symbols and four 4s as numbers...
But Pi can be gotten in certain ways

##lim_{x-> -1^{+}} =cos^{-1}(x)##

I'm not sure if cos^{-1} was defined exactluy at the value of (-1)

so that might be something like cos^{-4/4}(-4/4)= pi

It would appear so (and mine above could also be considered partially humour), but isn't solving an equation like inverting an operator or a function? Thus cosistent with:

And it is not limited by the video, or the original post [but since the OP sais now so we can now perhaps put it as a restriction ... (?)].

For example (aside/[or plus] the fact that functions can be considered as operarors, and operators are essentially functions):

If x is any real number (say π), and f any invertable real function (to avoid regular or perplexity form plain equations), and it happens that a number A which is directly/easily or else consructed by the 4 4s (say 4444) is related to x as A = f(x), then x = f^{-1}(A)
[e.g. x (or e.g. π) = f^{-1}(4444), and that's true for any [invertable] function f (providing it satisfies A = f(x) ) (e.g. a•x, sin, cos, exp, ln, log etc.) - just pick an appropriate one and be my guest! ...]

That's what I essentially did above, e.g.

where f(x) = (1/C)•x, A = 4444, and mfb did essentially the same with cos and arccos

since arccos = cos^{-1} ...

So are we joking or serious? I am a bit confused.

My opinion is that the problem on the video etc. is either trivial or one would have to put explicitly more restrictions, namely to the basic functions and operations only, with no arbitrary constants (other than the 4 4s and their obvious basic step derivatives).
[In other words functions e.g. like a•x, or a•cosx ... are not allowed, unless a=1 ... etc., otherwise one can do anything - it's trivial]

So the problem is not the variables but the arbitrary constants.

There is nothing inverse about it. The arccosine is a regular function (in more than one way). The cosine (restricted to [0,pi]) is the inverse function of the arccosine. And now?

But one has to define explicitly which functions are allowed to build from, and if inversing is allowed. (+see my edited posts above about arbitrariness of constants) But in any case I think your's is ok. I just wasn't sure if you were serious or not.