Algebra point of view on Tribonacci constant with nested radical

Question

Trying to find an expression in term of nested radical for the tribonacci constant we get a messy result starting with : $$S=\sqrt{1+S+\frac{1}{S}}$$

I have tried another expression like :

$$S=\sqrt{\frac{S+1}{S-1}}$$

But it reveals operation like "-" and the number $2$

So now I think it's impossible to find another expression just with "+" and the number $1$

Could it be prove using algebra or the definition says what it says ?

Thanks in advance !

Ps: I'm a beginner so if you can provide link to guide me it would be cool .