Let $\mathbb{Z}_p$ be the ring of p-adic integers. I want to show that the limit $\lim_na^{-(1+p+p^2+....+p^n)}$ exists in $\mathbb{Z}_p$, where $a \in \mathbb{Z}_p^{\times}$.
If we take the p-adic additive valuation $v_p(.)$ it would be $-\sum_{i=1}^{n} p^n$ which tends to $\infty$, thus the limit exists. It is trivial. Is there any flopy?
You forgot to say what kind of number $a$ is.
If $a \in 1 + p\mathbf Z_p$ then $a^k$ for $k \in \mathbf Z$ is $p$-adically uniformly continuous in $k$: for all integers $k$ and $\ell$, $$ |a^k - a^\ell|_p \leq |a-1|_p|k-\ell|_p $$ That's why the values can be $p$-adically interpolated to define $a^t$ for all $t \in \mathbf Z_p$, with values in $1+p\mathbf Z_p$: $a^t = \lim_{n \to \infty} a^{k_n}$ where $k_n$ is any sequence of integers such that $k_n \to t$ as $n \to \infty$.
In particular, since $1 + p + \cdots + p^n \to 1/(1-p)$ in $\mathbf Z_p$, $a^{-(1+p+\cdots+p^n)} \to a^{-1/(1-p)} = a^{1/(p-1)}$ as $n \to \infty$. This is the unique $x \in 1 + p\mathbf Z_p$ such that $x^{p-1} = a$. That the polynomial $x^{p-1} - a$ has a unique root in $\mathbf Z_p$ also follows from Hensel's lemma.
If $a \in \mathbf Z_p^\times$ and $a \not\in 1 + p\mathbf Z_p$ then that limit does not exist: it is the infinite product of the terms $a^{p^j}$, and if such a product exists then the terms $a^{p^j}$ tend to $1$, which forces $a \equiv 1 \bmod p$.