There are more ways of solving it depending on your circumstances.
The obvious way is through 2 views displays with different paths, or as parent-child
path3 for the first one and
path3/path9 for the second one (you can set them as Drupal tabs that way, so Drupal tab can play the role of your button "Show 6 more"). Set one to only show 3 items, the other one to show 9. Add a text area in the Views Footer of your first display and add the anchor HTML that leads to the page link of your second views display.
If you know a bit of Ajax you could display one of the displays as an attachment that has pager offset set to 3, and use Ajax button to make it display.
You can allow users to choose how many items are displayed by Exposing the Full or Mini pager with the option
Allow user to control the number of items displayed in this view and enter
3, 9 while setting
Items per page to 3.
This way you will see 3 items to start with, but there will be a dropdown above your results where users can choose either 3 or 9. If they choose 9 the page will reload to show 9 items in the results.
Using the Pager this way also gives you parameters on your links so you can even build links/button elsewhere on your site to link to a page with a particular number of results
The downside for you may be that there are also pager links at the bottom of the results (Previous, Next) that you might not want displayed. Maybe you can hide them with CSS.
You should also research into the Pager related modules, maybe Views flexible pager and Pagerer can help.