Does chemical compound with formula N₄O₁₀ exist?
PubChem database N4O10 (molecular formula) search shows that there is no known compound with this formula.

While $\ce{NO}$ (“mononitrogen monoxide”) and/or $\ce{NO2}$ (“mononitrogen dioxide”), having odd number of electrons, will dimerize or combine to form

$\ce{N2O2{: } 2NO -&gt; ON-NO}$
$\ce{N2O4{: } 2NO2 -&gt; O2N-NO2}$
$\ce{N2O3{: } NO + NO2 -&gt; ON-NO2}$

(note the $\ce{N-N}$ bond),

$\ce{N2O5}$ is formed by the condensation of nitric acid $\ce{HNO3}$:

$\ce{ 2 { }O2N-OH -&gt;[-H2O] O2N-O-NO2 }$  

(note the $\ce{N-O-N}$ bond), thus can be considered nitric acid anhydride. And unlike polyphosphoric acid formation from $\ce{H3PO4}$, there are no remaining hydroxy groups for further condensation.

With non-standard formal approach, $\ce{N2O5}$ already is a dimer of hypothetical franctional-oxygen-atom-count “mononitrogen(V) oxide” $\ce{NO_{2.5}}$.