By repeatedly replacing the value of x with the formula on the left side, x converges rapidly.  Starting with a guess of x=2, it converges thus:

2.300761708156168
2.302761656377654
2.302775540639462
2.302775637057741
2.302775637727312
2.302775637731962
2.302775637731994
2.302775637731995
2.302775637731995
...

I squared the terminal number to see if this was the square root of an integer or of a rational number, which it was not, but it was exactly 3 higher than the number itself (it came out to 5.302775637731996, the difference in the final digit attributable to rounding differences).

So, solving x^2-3 = x:

x^2 - x - 3 = 0

x = (1 +/- sqrt(1 + 12)) / 2

As the number is positive, we get (1 + sqrt(13)) / 2.

The program:

DEFDBL A-Z
x = 2
FOR i = 1 TO 40
  x = SQR(4 + SQR(4 - SQR(4 + SQR(4 - x))))
  PRINT x
NEXT