Retrieve matrix content from Entries form field in an Entry
When you loop through your matrix, you're missing the s on components. Should be:

{% for block in template.components %}


Each item in that loop will be one of the blocks in the matrix. So then you can call {{ block.component }} directly.

FYI, in your code, it looks like that component field in the matrix block is loopable as well? If you're trying to loop through an entries field, and then loop through a matrix field per entry, and then loop through another field per matrix block, expect some serious performance issues.

Full code reference: 

{% for template in entry.chooseTemplateExample %}
  &lt;a href="{{ template.url }}" class="btn btn-primary"&gt;View {{ template.title }} example&lt;/a&gt;&lt;br&gt;

  {% for block in template.components %}
    {% for component in block.component %} {!-- Is this field really loopable? --}
      {{ title }}
    {% endfor %}
  {% endfor %}
{% endfor %}