Highly recursive functions that majored primitive recursive functions have already been invented by Ackerman in the late 1920s. In 1970s Takeuchi came up with a highly recursive function that has astonishing simple closed form solution. The recursive function is:
 Y for X <= Y
tarai(X,Y,Z) := 
 tarai(tarai(x1,y,z),tarai(y1,z,x),tarai(z1,x,y))
And its close form:
 Y for X <= Y

tarai(X,Y,Z) :=  Z for Y <= Z

 X
But McCarthy spoiled it all by a slight modification. Instead of returning Y in the first case of the recursive definition, in his version the value of Z is returned. This function has to become known as the tak function.
One test iteration will compute the tak function for the arguments (18, 12, 6).