How to put a search box way up above your page

Look up. Way up above the site title.

Page settings (click the gear)

Page settings (click the gear)

There should be a search bar there (if there isn't, see the warning below and let me know). I got it by going to this page's settings  as shown here.

Then I inserted some HTML on the "Advanced" tab under the page settings.

Inject some custom HTML in the page's header

Inject some custom HTML in the page's header

I got it by copying the HTML from another page with a SquareSpace Search on it. Mine reads:

<div class="sqs-search-ui-text-input sqs-search-ui-button-wrapper color-dark" 
     data-source="block" 
     data-preview="true" 
     data-collection="" 
     id="yui_3_17_2_1_1460597366677_168" 
     style="margin:30px auto;max-width:1100px;">
  <div class="spinner-wrapper" 
       id="yui_3_17_2_1_1460597366677_293" 
       hidden="hidden" 
       style="display: none;">
  </div>
  <input type="search" 
         class="search-input hover-effect" 
         value="" 
         placeholder="Search" 
         id="yui_3_17_2_1_1460597366677_337">
  <div id="yui_3_17_2_1_1460597366677_173" 
       class="yui3-widget sqs-widget sqs-search-preview">
    <div id="yui_3_17_2_1_1460597366677_175" 
         class="sqs-search-preview-content">
    </div>
  </div>
  <div id="yui_3_17_2_1_1460600921866_173" 
       class="yui3-widget sqs-widget sqs-search-preview">
    <div id="yui_3_17_2_1_1460600921866_175" 
         class="sqs-search-preview-content">
    </div>
  </div>
  <div id="yui_3_17_2_1_1460600921866_314" 
       class="yui3-widget sqs-widget sqs-search-preview">
    <div id="yui_3_17_2_1_1460600921866_316" 
         class="sqs-search-preview-content">
    </div>
  </div>
</div>

A word of warning

This isn't really the right way to do any of this. You should probably find a template that has a search you like or enable developer mode and do it properly. I just wanted to see if I could do it. When SquareSpace updates their templates it will break. I know that because it has done it before.