The idea in mixing is to get good balance and sound quality in all the dimensions of the mix. "As loud as possible" isn't necessary. If you have noise problems you should look at how you're doing your tracking.
This is good advice.
If you are having noticable noise problems when you boost volume during mixing, then chances are you haven't recorded the tracks at the right level to begin with. In other words, your signal is recorded too close to the noise floor.
The first thing to do is to troubleshoot your recording setup and get the noise levels down as far as possible. This means tracking down any hums, hisses, or other audio oddities, and solving them.
If you are using a mic, you need to find the optimum settings for your mic and preamp. All gear has a sweet spot, and you need to experiment in order to find that. If you are using budget gear, chances are you are setting it up outside the sweet spot.
I mention budget gear only because low priced gear tends to have a very narrow sweet spot. Sometimes it can be a challenge to find it and make it usable.
Once you've sorted out the various issues with your recording setup, then you probably need to record at hotter levels. This gets your usable signal further away from the noise floor, ultimately allowing you to mix with quiter background noise.
Finally, the point of mixing is *not* to get the signal as loud as possible without clipping. That's not a factor at all, and I mean zero. It's as XLR said: "The idea in mixing is to get good balance and sound quality in all the dimensions of the mix."
In fact, when mixing your peaks should not go over -6 at the very highest. Keeping it anywhere from -12 to -6 will be a very good level, and ultimately will sound better.
Once you are finished mixing, then you can boost levels in the mastering process. But striving for ear shattering levels while mixing is not ideal at all. If it doesn't sound loud enough to you, just turn up your monitors.