Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)^{\frac{1}{50}}.$$ If the function $$g(x)=f(f(f(x)))+f(f(x)),$$ then the greatest integer less than or equal to $$g(1)$$ is ____________.

Given,

$$f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)^{\frac{1}{50}}$$

Simplifying,

$$f(x)=\left((2-x^{25})(2+x^{25})\right)^{\frac{1}{50}}$$

$$f(x)=\left(4-x^{50}\right)^{\frac{1}{50}}$$

Now,

$$f(f(x))=\left(4-\left((4-x^{50})^{\frac{1}{50}}\right)^{50}\right)^{\frac{1}{50}}$$

$$=\left(4-(4-x^{50})\right)^{\frac{1}{50}}$$

$$=(x^{50})^{\frac{1}{50}}=x$$

Hence,

$$f(f(x))=x$$

Given,

$$g(x)=f(f(f(x)))+f(f(x))$$

Using $$f(f(x))=x,$$

$$g(x)=f(x)+x$$

Now,

$$f(1)=\left(4-1^{50}\right)^{\frac{1}{50}}=3^{\frac{1}{50}}$$

Therefore,

$$g(1)=3^{\frac{1}{50}}+1$$

Since,

$$1<3^{\frac{1}{50}}<2$$

we get,

$$2<g(1)<3$$

Therefore, the greatest integer less than or equal to $$g(1)$$ is $$\boxed{2}$$.