Let $G{}$ be an arbitrary finite group, and let $t_n(G)$ be the number of functions of the form \[f:G^n\to G,\quad f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n\quad(a_0,\ldots,a_n\in G).\]Determine the limit of $t_n(G)^{1/n}$ as $n{}$ tends to infinity.
1. **Define the problem and the function count**: We are given a finite group \( G \) and need to determine the limit of \( t_n(G)^{1/n} \) as \( n \) tends to infinity, where \( t_n(G) \) is the number of functions of the form:
   \[
   f: G^n \to G, \quad f(x_1, x_2, \ldots, x_n) = a_0 x_1 a_1 \cdots x_n a_n \quad (a_0, \ldots, a_n \in G).
   \]

2. **Claim and action definition**: We claim that \( t_n(G) = \frac{|G|^{n+1}}{|Z(G)|^n} \), where \( Z(G) \) is the center of \( G \). To prove this, consider the action of \( G^{n+1} \) on the set of such functions by:
   \[
   ((g_0, g_1, g_2, \ldots, g_n)f)(x_1, x_2, \ldots, x_n) = g_0 f(x_1 g_1, x_2 g_2, \ldots, x_n g_n).
   \]

3. **Transitivity and stabilizer**: This action is transitive. To find the number of functions, we study the stabilizer of the element \( f_0(x_1, x_2, \ldots, x_n) = x_1 x_2 \cdots x_n \) under this action. If \( (c_0, c_1, \ldots, c_n) \) is an element of the stabilizer, then for any \( n \)-tuple of elements of \( G \), the relationship:
   \[
   x_1 x_2 \cdots x_n = c_0 x_1 c_1 \cdots x_n c_n
   \]
   must hold.

4. **Fixing elements and center**: By fixing all but the first element, we get \( x_1^{-1} c_0 x_1 \) constant for all \( x_1 \), implying \( c_0 \in Z(G) \). This also implies:
   \[
   c_0 x_1 c_1 \cdots x_n c_n = x_1 c_0 c_1 \cdots x_n c_n,
   \]
   so by induction, we get \( c_0 c_1 \cdots c_k \in Z(G) \) for all \( k \), which is equivalent to \( c_k \in Z(G) \) for all \( k \).

5. **Stabilizer subgroup**: Moreover, for the two functions to be equal, we also need \( c_0 c_1 \cdots c_n = e \). Conversely, any such \( (n+1) \)-tuple stabilizes \( f_0 \), so the stabilizer is the subgroup of \( Z(G)^{n+1} \) where the product of all coordinates is the identity, which has order \( |Z(G)|^n \).

6. **Counting functions**: Therefore, the number of functions is:
   \[
   t_n(G) = \frac{|G|^{n+1}}{|Z(G)|^n}.
   \]

7. **Limit calculation**: We now calculate the limit:
   \[
   \lim_{n \to \infty} t_n(G)^{1/n} = \lim_{n \to \infty} \left( \frac{|G|^{n+1}}{|Z(G)|^n} \right)^{1/n} = \lim_{n \to \infty} \left( |G|^{n+1} \right)^{1/n} \left( |Z(G)|^{-n} \right)^{1/n}.
   \]
   Simplifying, we get:
   \[
   \lim_{n \to \infty} |G|^{(n+1)/n} |Z(G)|^{-1} = |G| \cdot |Z(G)|^{-1} = \frac{|G|}{|Z(G)|}.
   \]

The final answer is \(\boxed{\frac{|G|}{|Z(G)|}}\).