Partial answers: power series expansions around 0 and $\infty$
Define for $k\in\mathbb{Z}, k\ge 0$
$$ f_k(x) = \sqrt{x^k+\sqrt{x^{k+1}+\sqrt{x^{k+2}+\ldots}}} = \sqrt{x^k+f_{k+1}(x)}$$
where the meaning of the infinite square root iteration will be made precise below. The function of the OP is then $f_1(x)$.
For the power series expansion at 0, let us assume in all what follows that $0\lt x\ll 1$. Then for $k>1$, the term $f_{k+1}(x)=\sqrt{x^{k+1}+f_{k+2}(x)}$ dominates $x^k$ in $\sqrt{x^k+f_{k+1}(x)}$, simply because $x^k\ll x^\frac{k+1}{2}$, but $x^{k+1}$ is again dominated by $f_{k+2}(x)$ and so on, so we have approximately $f_k(x)\approx\sqrt{f_{k+1}(x)}\approx\sqrt{\sqrt{f_{k+2}(x)}}\ldots$ But taking square roots again and again on a positive value will aproach 1. Thus it is sensible to conjecture that for $0\lt x\ll 1$, we have $f_k(x)\approx 1$.
Replacing $f_{m+1}(x)$ with 1 somewhere inside the infinitely iterated expressions leads to the following partial expressions ($k,m\in\mathbb{Z}, 0\le k\le m$):
$$f_{k,m}(x) := \sqrt{x^k+\sqrt{x^{k+1}+\sqrt{x^{k+2}+\ldots\sqrt{x^m+1}}}},$$
and we can define
$$f_k(x):=\lim_{m\to\infty}f_{k,m}(x)$$
if that limit exists (which I won't elaborate). Now it is easily seen that $f_{k,m}(x) = 1 + \mathcal{O}(x^k)$ (just start from $f_{m,m}(x) =\sqrt{x^m+1}= 1 + \mathcal{O}(x^m)$ and proceed by induction on decending $k$: $f_{k,m}(x)=\sqrt{x^k+f_{k+1,m}(x)}=\sqrt{x^k+1+\mathcal{O}(x^{k+1})}=1 + \mathcal{O}(x^k)$).
But this means that for any $m'>m$
\begin{align}
f_{k,m'}(x) & = \sqrt{x^k+\sqrt{x^{k+1}+\sqrt{x^{k+2}+\ldots\sqrt{x^m+f_{m+1,m'}(x)}}}}\\
& = \sqrt{x^k+\sqrt{x^{k+1}+\sqrt{x^{k+2}+\ldots\sqrt{x^m+1+\mathcal{O}(x^{m+1})}}}}\\
& = f_{k,m}(x) + \mathcal{O}(x^{m+1})
\end{align}
and so
$$f_k(x)=\lim_{m'\to\infty}f_{k,m'}(x)=f_{k,m}(x) + \mathcal{O}(x^{m+1}).$$ Hence, the terms of the power series expansion of $f_k(x)$ up to order $x^m$ are determined by the power series expansion of $f_{k,m}(x)$. For example, the result for $f_1(x)$ up to order $x^{20}$ reads
$$f_1(x)= 1 + \frac{1}{2}x + \frac{1}{8}x^2 + \frac{1}{16}x^3 - \frac{5}{128} x^4
- \frac{5}{256} x^5 - \frac{19}{1024} x^6 + \frac{13}{2048} x^7
- \frac{397}{32768} x^8 + \frac{243}{65536} x^9 + \frac{79}{262144} x^{10}
+ \frac{6415}{524288} x^{11} + \frac{10959}{4194304} x^{12}
- \frac{6321}{8388608} x^{13} - \frac{283323}{33554432} x^{14}
+ \frac{171429}{67108864} x^{15} + \frac{4224323}{2147483648} x^{16}
+ \frac{22138947}{4294967296} x^{17} - \frac{25215333}{17179869184}x^{18}
- \frac{83594725}{34359738368}x^{19} - \frac{1538702507}{274877906944}x^{20}
+ \mathcal{O}(x^{21}).$$
For analyzing the behavior of $f_k(x)$ for large positive $x$, consider
\begin{align}
f_k(x^{-2})&=\sqrt{x^{-2k}+\sqrt{x^{-2k-2}+\sqrt{x^{-2k-4}+\ldots}}}\\
&=x^{-k}\sqrt{1+x^{k-1}\sqrt{1+x^k\sqrt{1+x^{k+1}\sqrt{1+\ldots}}}}\\
&=x^{-k}g_{k-1}(x),
\end{align}
again for $0<x\ll 1$, where
$$g_k(x)= \sqrt{1+x^{k}\sqrt{1+x^{k+1}\sqrt{1+x^{k+2}\sqrt{1+\ldots}}}}.$$
Setting for $k\le m$
$$g_{k,m}(x):= \sqrt{1+x^{k}\sqrt{1+x^{k+1}\sqrt{1+\ldots x^{m-1}\sqrt{1+x^m}}}},$$
and we can define
$$g_k(x):=\lim_{m\to\infty}g_{k,m}(x)$$
and we find $g_{k,m}(x)=1+\mathcal{O}(x^k)$ and for any $m'> m$
\begin{align}
g_{k,m'}(x) & = \sqrt{1+x^k\sqrt{1+x^{k+1}\sqrt{1+\ldots x^{m-1}\sqrt{1+x^m g_{m+1,m'}(x)}}}}\\
& = \sqrt{1+x^k\sqrt{1+x^{k+1}\sqrt{1+\ldots x^{m-1}\sqrt{1+x^m(1+\mathcal{O}(x^{m+1})}}}}\\
& = g_{k,m}(x) + \mathcal{O}(x^{(k+m+1)(m+2-k)/2}),
\end{align}
so again, the power series expansion of $g_k(x)$ up to any given order can be determined by the power series expansion of $g_{k,m}(x)$ for sufficiently large $m$.
For instance, for determining $g_1(x)$ up to order $x^{20}$, expanding $g_{1,5}(x)$ is sufficient, yielding:
$$g_1(x)=1 + \frac{1}{2} x - \frac{1}{8} x^2 + \frac{5}{16} x^3
- \frac{21}{128} x^4 + \frac{15}{256} x^5 + \frac{27}{1024} x^6
+ \frac{157}{2048} x^7 - \frac{4237}{32768} x^8 + \frac{1627}{65536} x^9
+ \frac{15585}{262144} x^{10} + \frac{20179}{524288} x^{11}
- \frac{420737}{4194304} x^{12} + \frac{136155}{8388608} x^{13}
+ \frac{606675}{33554432} x^{14} + \frac{3116173}{67108864} x^{15}
- \frac{166576957}{2147483648} x^{16} + \frac{258982675}{4294967296} x^{17}
- \frac{117088187}{17179869184} x^{18} - \frac{516645801}{34359738368} x^{19}
- \frac{23704687899}{274877906944} x^{20} + \mathcal{O}(x^{21})$$
and
$$g_0(x)=\sqrt{1+g_1(x)}=\sqrt{2}\left(1 + \frac{1}{8} x - \frac{5}{128} x^2 + \frac{85}{1024} x^3 - \frac{1709}{32768} x^4 + \frac{6399}{262144} x^5
- \frac{8145}{4194304} x^6 + \frac{828477}{33554432} x^7 - \frac{83481725}{2147483648} x^8
+ \frac{231319419}{17179869184} x^9 + \frac{2532368405}{274877906944} x^{10}
+ \frac{29815364515}{2199023255552} x^{11} - \frac{2122499603177}{70368744177664} x^{12}
+ \frac{5230968689963}{562949953421312} x^{13} + \frac{7443547207831}{9007199254740992} x^{14}
+ \frac{1141411701025037}{72057594037927936} x^{15}
- \frac{231372106336231965}{9223372036854775808} x^{16}
+ \frac{1498156069006490195}{73786976294838206464} x^{17}
- \frac{8082528897875176135}{1180591620717411303424} x^{18}
+ \frac{18359172053830212871}{9444732965739290427392} x^{19}
- \frac{8183042653064552822819}{302231454903657293676544} x^{20} + \mathcal{O}(x^{21})\right).$$
This yields immediately the behavior of $f_1(x)=\sqrt{x}g_0(1/\sqrt{x})$ for large $x$:
$$f_1(x)=\sqrt{2x}\left(1 + \frac{1}{8\sqrt{x}}- \frac{5}{128x} + \frac{85}{1024 \sqrt{x^3}} - \frac{1709}{32768 x^2} + \frac{6399}{262144\sqrt{x^5}} - \ldots\right).$$
Interestingly, the recursion
$$g_k(x)^r=\left(1+x^kg_{k+1}(x)\right)^{r/2}=\sum_{a=0}^\infty{\frac{r}{2}\choose a}x^{ak}g_{k+1}(x)^a$$
can be used to get for $k\ge 1, r\ge 0$ the expression
$$g_k(x)^r=\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\sum_{a_3=0}^\infty\ldots
{\frac{r}{2}\choose a_1}{\frac{a_1}{2}\choose a_2}{\frac{a_2}{2}\choose a_3}\ldots
x^{a_1 k + a_2(k+1)+a_3(k+2)+\ldots},$$
such that the coefficients of
$$g_1(x)=\sum_{r=0}^\infty c_rx^r$$
can be written as
$$c_r=\sum_{a_1}\sum_{a_2}\sum_{a_3}\ldots{\frac{1}{2}\choose a_1}{\frac{a_1}{2}\choose a_2}{\frac{a_2}{2}\choose a_3}\ldots,$$
where, for fixed $r$, the summation variables are restricted to $a_i\ge 0$ and $\sum_iia_i=r$, such that the sum is in fact finite. And because the binomial coefficients ${0\choose a}$ are zero for $a>0$, and more generally ${a_i/2\choose a_{i+1}}=0$ for $a_i$ even and $a_{i+1}>a_i/2$, the terms of the sum are non-zero only for those values $(a_1, a_2,\ldots)$ where for each even $a_i$ holds $a_{i+1}\le\frac{a_i}{2}$.
The power expansions suggest that there is no simple expression for the function of the OP. But this does not exclude the possibility that it might be an algebraic function in the sense that there might be a polynomial $p(x,y)$ in two variables $x$ and $y$, such that $p(x,f_1(x))=0$.
