What exactly is the fixed field of the map $t\mapsto t+1$ in $k(t)$?

Suppose first that $k$ is algebraically closed and of characteristic zero.

Let $P$, $Q$ be coprime. The equality $P(t)/Q(t)=P(t+1)/Q(t+1)$ implies that $P(t)$ and $P(t+1)$ have the same zeroes in $k$, and the same for $Q(t)$ and $Q(t+1)$. This is easily seen to be impossible, unless they are constant.

If now $k$ is not necessarily algebraically closed and $\overline k$ is its algebraic closure, the invariants in $k(t)$ are contained in the invariants of $\overline k(t)$. Therefore again in this case we only have constants.

Now, if $k$ is of positive characteristic $p$, things are more subtle. Let $\sigma:k[t]\to k[t]$ be the unique autmorphism such that $\sigma(t)=t+1$, and let $\sigma:k(t)\to k(t)$ be its natural extension. The difference now is that $\sigma $ has order $p$.

If $f/g\in k(t)$, then $$ \frac fg = \frac{f\sigma(g)\sigma^2(g)\cdots\sigma^{p-1}(g)}{g\sigma(g)\sigma^2(g)\cdots\sigma^{p-1}(g)} $$ and the denominator in the right is $\sigma$-invariant. We thus see that every element in $k(t)$ can be written with an invariant denominator, so it is itself invariant iff its numerator is invariant. This means that $k(t)^\sigma$ is the quotient field of the ring of $\sigma$-invariant polynomials $k[t]^\sigma$.

Can you describe the elements of $k[t]^\sigma$?

Later: Suppose $f$ is in $k[t]^\sigma$ and, at first, that $k$ is algebraically closed. If $\alpha$ is a root of $f$, then $\alpha+1$, $\dots$, $\alpha+p-1$ are also roots of $f$ because of invariant, and therefore the product $\phi_a=\prod_{i=0}^{p-1}(t-a-i)=t^p-t-a^p+a$ divides $f$. The quotient $f/\phi_a$ is also in $k[t]^\sigma$. By induction, we conclude that $f$ is a product of polynomials of the form $\phi_a$ for various $a$s in $k$.

In characteristic $p$ case we can also proceed as follows. We get from Professor Suárez-Alvarez' answer that $u=t^p-t$ is contained in the fixed field $L=k(t)^\sigma$, so $k(u)\subseteq L$. Galois theory tells us that $[k(t):L]=\mathop{ord}(\sigma)=p$. OTOH clearly $[k(t):k(u)]=[k(t,u):k(u)]\le p$, because the minimal polynomial $m(x)$ of $t$ over $k(u)$ must be a factor of $x^p-x-u$. Therefore we can conclude that $L=k(u)$, and also that $m(x)=x^p-x-u$.

We can also tell right away that the fixed field of a finite group of automorphisms of $k(t)$ will be of the form $k(v)$ for some rational function $v$. This follows from Lüroth's theorem.