Nothing is impossible. Here’s a solution that simply sets the value of a text input whenever the value of the <select> changes (rendering has been tested on Firefox and Google Chrome):
.select-editable {position:relative; background-color:white; border:solid grey 1px; width:120px; height:18px;}
.select-editable select {position:absolute; top:0px; left:0px; font-size:14px; border:none; width:120px; margin:0;}
.select-editable input {position:absolute; top:0px; left:0px; width:100px; padding:1px; font-size:12px; border:none;}
.select-editable select:focus, .select-editable input:focus {outline:none;}<div class="select-editable">
<select onchange="this.nextElementSibling.value=this.value">
<option value=""></option>
<option value="115x175 mm">115x175 mm</option>
<option value="120x160 mm">120x160 mm</option>
<option value="120x287 mm">120x287 mm</option>
</select>
<input type="text" name="format" value=""/>
</div>jsfiddle: https://jsfiddle.net/nwH8A/
The next example adds the user input to the empty option slot of the <select> (thanks to @TomerPeled). It also has a little bit more flexible/variable CSS:
.select-editable {position:relative; width:120px;}
.select-editable > * {position:absolute; top:0; left:0; box-sizing:border-box; outline:none;}
.select-editable select {width:100%;}
.select-editable input {width:calc(100% - 20px); margin:1px; border:none; text-overflow:ellipsis;}<div class="select-editable">
<select onchange="this.nextElementSibling.value=this.value">
<option value=""></option>
<option value="115x175 mm">115x175 mm</option>
<option value="120x160 mm">120x160 mm</option>
<option value="120x287 mm">120x287 mm</option>
</select>
<input type="text" oninput="this.previousElementSibling.options[0].value=this.value; this.previousElementSibling.options[0].innerHTML=this.value" onchange="this.previousElementSibling.selectedIndex=0" value="" />
</div>jsfiddle: https://jsfiddle.net/pu7cndLv/1/
DataList
In HTML5 you can also do this with the <input> list attribute and <datalist> element:
<input list="browsers" name="browser">
<datalist id="browsers">
<option value="Internet Explorer">
<option value="Firefox">
<option value="Chrome">
<option value="Opera">
<option value="Safari">
</datalist>
(click once to focus and edit, click again to see option dropdown)jsfiddle: https://jsfiddle.net/hrkxebtw/
But this acts more like an auto-complete list; once you start typing, only the options that contain the typed string are left as suggestions. Depending on what you want to use it for, this may or may not be practical.