  Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there
exist infinitely many Galois representations
$\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$
which are unramified outside $\{p, \infty\}$ with large image. More precisely,
the Galois representations constructed have image containing the kernel of the
mod-$p^t$ reduction map $SL_n(\mathbb{Z}_p)\rightarrow
SL_n(\mathbb{Z}/p^t\mathbb{Z})$, where $t:=8(n^2-n)\left(3+\lfloor
log_p(2^n+1)\rfloor\right)+8$. The results are proven via a purely Galois
theoretic lifting construction. When $p\equiv 1\mod{4}$, our results are
conditional since in this case, we assume a very weak version of Vandiver's
conjecture.
