How to fix the problem of getting error dimension too large or the File ended while scanning use of pgfplots@@environment@axis?

You have several problems in your code. Indeed, you open environments or braces and do not close them. For example, here, you begin center environment without ending it. But the error you get from pgfplots comes from the fact that you open a brace after every extra x tick/.style without closing it. Then, the dimension too large comes from the fact that the functions you are trying to compute are indeed becoming too large for pgf to compute when x is close to 0 since they tend to infinity, and you even try to make pgf compute 1/0 with the domain you define.

A way to avoid that would be to split your graphs in two parts, one stopping somewhere before 0 and the other starting after 0. Another solution is to use the key restrict y to domain in the axis environment to filter out the y values when they become too large.

A functioning example with the restriction key is:

documentclass{standalone}

usepackage{pgf,tikz,pgfplots}
    usetikzlibrary{arrows}
    pgfplotsset{compat=1.15}
    
usepackage{mathrsfs}

begin{document}
    
    begin{tikzpicture}
    
        begin{axis}[%
            width = 2.5in,
            ymin = -1.5,
            xmin = -3.5,
            ymax = 3.5,
            xmax = 5,
            xtick = empty,
            extra x ticks = {3.14, 6.28},
            xticklabel style = {%
                anchor = north east
            },
            ytick = {-1,0,1,2,3},
            axis lines = center,
            xlabel = $x$,
            ylabel = $y$,
            domain = -3.5:4.8,
            restrict y to domain = -1.5:5,
            samples = 200,
            axis line style = thick,
        ]
        
            addplot [thick] {1/(x^2)};
            
            addplot [black, thick] {1/(x^4)};
            
            addplot [black, thick] {1/(x^6)};
            
            node[below right] at (axis cs: 0, 0) {$O$};
            
        end{axis}
        
    end{tikzpicture}
    
end{document}

which yields:

As a side note, if you declare a domain key in the axis environment options, it's no use declaring it again in the addplot options unless it is different.