<p>Our site has the default Stack Exchange logo (text balloon with text 3D), is it possible to change this logo?</p>
<p>I was thinking of something like this:</p>
<p><a href="https://i.stack.imgur.com/n4VXE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/n4VXE.png" alt="enter image description here" /></a></p>
<p>or</p>
<p><a href="https://i.stack.imgur.com/w05Jd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w05Jd.png" alt="enter image description here" /></a></p>
<p>If it is possible, we could try to write out a competition and vote! E.g. instead of the lines, the printed image of 3D</p>

Yes, we can! Or at least discuss if we like a different logo.
From
this answer
on Meta Stack Exchange question "
What's the process to change a site logo?
" we can read:
If you have an issue with a logo on a site, the best place to start is to open a discussion on that site's meta. Tag it
discussion
and
design
, and see what the overall community feeling is.
Another part reads:
Community managers
monitor per-site metas, so if/when the discussion concludes and the site's community largely supports a change, they can bring your concerns to the design team.
There is no specific mention if Beta state sites can modify the logo, Beta sites share the smae layout set out by the Stack Exchange designers:
Our designers come up with the overall site design (including the logo), with some input from the communities. (With the exception of sites that are still in beta or have only recently graduated - those all share the same design.)
However, we can start a discussion whether we would like to have a different logo/favicon. Feel free to add your thoughts ans an aswer to the question.