Let \( \Large x=2+\frac{1}{2+\frac{1}{2+...\infty}} \)

=> \( \Large x = 2+\frac{1}{x} => x^{2}-2x-1=0 \)

=> \( \Large x=\frac{2\pm \sqrt{4+4}}{2} \)

=> \( \Large x=1\pm \sqrt{2} \)

But the value of given expression cannot be negative or less than 2, therefore \( \Large 1+\sqrt{2} \) is required answer.