#Javascript ES6 (REPL), 52 bytes
var a=a||'';$=_=>a+=$=${$};$();.repeat(a?1:2);$();
Initial working implementation, may be golfable.
y.onclick=_=>z.innerText=eval("var a=a||'';$=_=>a+=`$=${$};$();`.repeat(a?1:2);$();".repeat(x.value))<input id=x><button id=y>Submit!</button><pre id=z></pre>#Explanation
Based on my usual quine implementation ($=_=>`$=${$};$()`;$()).
var a||'' coerces a to an empty string if it isn't defined; otherwise, a is left as is. a will store our final result.
Each function call, a will be checked to see if it is empty/falsy. The first function call will append 2 copies of the quine because a is empty; subsequent calls will only append 1 copy because a is no longer empty.
The last function call will output the final value of a after finishing.