I think you're gonna want to add the assumption that $x_{n}$ grows unboundedly. Even with the positive coefficient condition that Darij suggested, you have examples like $x_4=\frac{x_1+x_2}{2x_3}$ that give the constant sequence $1,1,1,\ldots$.

(By the way, in class Pasha did suggest that in all "known" or "studied" examples of integrability/non-integrability it suffices to check what happens when you plug in all $1$'s for the initial variables.)

EDIT:

Actually, how about $x_4 = \frac{x_1+x_2}{x_3}$?

To me with $x_1=x_2=x_3 =1$ this gives the sequence $1,1,1,2,1,3,1,4,1,5,...$ but it certainly is not Laurent because for instance we have $x_5=\frac{x_2x_3+x_3^2}{x_1+x_2}$.

Consider $x_{n+3} = \frac{x_n+x_{n+1}}{x_{n+2}}$.

With $x_1=x_2=x_3 =1$ this gives the sequence $1,1,1,2,1,3,1,4,1,5,...$ but it certainly is not Laurent because for instance we have $x_5=\frac{x_2x_3+x_3^2}{x_1+x_2}$.

I think you're gonna want to add the assumption that $x_{n}$ grows unboundedly. Even with the positive coefficient condition that Darij suggested, you have examples like $x_4=\frac{x_1+x_2}{2x_3}$ that give the constant sequence $1,1,1,\ldots$.

(By the way, in class Pasha did suggest that in all "known" or "studied" examples of integrability/non-integrability it suffices to check what happens when you plug in all $1$'s for the initial variables.)

I think you're gonna want to add the assumption that $x_{n}$ grows unboundedly. Even with the positive coefficient condition that Darij suggested, you have examples like $x_4=\frac{x_1+x_2}{2x_3}$ that give the constant sequence $1,1,1,\ldots$.

(By the way, in class Pasha did suggest that in all "known" or "studied" examples of integrability/non-integrability it suffices to check what happens when you plug in all $1$'s for the initial variables.)

EDIT:

Actually, how about $x_4 = \frac{x_1+x_2}{x_3}$?

To me with $x_1=x_2=x_3 =1$ this gives the sequence $1,1,1,2,1,3,1,4,1,5,...$ but it certainly is not Laurent because for instance we have $x_5=\frac{x_2x_3+x_3^2}{x_1+x_2}$.