Is there an algorithm to define a recursive function such that consecutive terms approach any arbitrary constant?
If $\alpha$ is real algebraic and a root of the polynomial with rational coefficients $$p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ and we have $|\beta|<\alpha$ for all other (real or complex) roots $\beta$ of $p$, then for a sequence defined by the recursion $$x_{k}=-a_{n-1}x_{k-1}-\cdots-a_1x_{k-n+1}-a_0x_{k-n}$$ (and almost any choice of initial values), the quotients $\frac {x_{k+1}}{x_k}$ will converge to €$\alpha$.
For algebraic numbers that are not maximal in the above sense, you are out of luck.