#JavaScript, (114) 109 bytes

This is my first entry in a codegolf contest! Please tell me how to improve.

    x="repeat",w="$$\\varphi=";y=n=>{return a=!n?w+"1+\\dots$$":w+"1+\\cfrac1{"[x](n)+"1+\\ddots"+"}"[x](n)+"$$"}

Previous example:

    x="repeat";y=n=>{return a=!n?"$$\\varphi=1+\\dots$$":"$$\\varphi="+"1+\\cfrac1{"[x](n)+"1+\\ddots"+"}"[x](n)+"$$"}

Paste into browser console and call as `f(n)` where `n` is the number of 'steps'.

**Simplified code**:
<!-- language: lang-js -->

    var x = "repeat";
    function y(n) {
       if(n === 0) {
          return "$$\\varphi=1+\\dots$$";
       } else {
          return "$$\\varphi=" + "1+\\cfrac1{".repeat(n) + "1+\\ddots"+"}".repeat(n)+"$$";
       } //                                   ^in golfed version, String.repeat(n) 
        //                                     is called as String[x](n)
    }

**Returns**:

 - $n=0$: `$$\varphi=1+\dots$$` which renders as:
$$\varphi=1+\dots$$
 - $n=1$: `$$\varphi=1+\cfrac1{1+\ddots}$$` which renders as:
$$\varphi=1+\cfrac1{1+\ddots}$$
 - $n=2$: `$$\varphi=1+\cfrac1{1+\cfrac1{1+\ddots}}$$` which renders as:
$$\varphi=1+\cfrac1{1+\cfrac1{1+\ddots}}$$
 - $n=3$: `$$\varphi=1+\cfrac1{1+\cfrac1{1+\cfrac1{1+\ddots}}}$$` which renders as:
$$\varphi=1+\cfrac1{1+\cfrac1{1+\cfrac1{1+\ddots}}}$$