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In this paper, we investigate nonlinear fractional differential equations of arbitrary order with advanced arguments \begin{equation*}\left\{\begin {array}{ll} D^\alpha_{0^+} u(t) +a(t)f(u(\theta(t)))=0,&0<t<1,~n-1<\alpha\le n,\\ u^{(i)}(0)=0,&i=0,1,2,\cdots,n-2,\\ ~[D^\beta_{0^+} u(t)]_{t=1}=0,&1\le \beta\le n-2, \end {array}\right.\end{equation*} where $n>3\,\, (n\in\mathbb{N}),~D^\alpha_{0^+}$ is the standard Riemann-Liouville fractional derivative of order $\alpha,$ $f: [0,\infty)\to [0,\infty),$ $a: [0,1]\to (0,\infty)$ and $\theta: (0,1)\to (0,1]$ are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established.