Solution:

Working from the outside in, $\dfrac{144}{53}-2=\dfrac{38}{53},$ so the inner fraction equals $\dfrac{38}{53}.$

Its reciprocal gives $1+\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{53}{38},$ so $\cfrac{1}{2+\frac{2}{3+x}}=\dfrac{15}{38}.$

Then $2+\dfrac{2}{3+x}=\dfrac{38}{15},$ so $\dfrac{2}{3+x}=\dfrac{8}{15},$ giving $3+x=\dfrac{15}{4}.$

Therefore $x=\dfrac{15}{4}-3=\dfrac{3}{4}.$

Thus, the correct answer is A.