The short answer is: you can't. Or at least not with the way things currently work with Gmail. As you've probably noticed, your only options are filtering by a tag or using the archive to hide things. A glance at the advanced search tools show what you are after isn't really accommodated.
Your question is in a similar vein to those asking how to find unlabeled messages, and a similar answer applies. You can try and build a search that explicitly excludes every other label, e.g.:
label:inbox -label:{labela labelb labelc}
You can build the search once then save the URL, but given you mention that you have a ton of labels I know it is far from ideal. If you have a look at the answersanswers to questions on finding unlabeled messages you'll see you're not alone.