Chamonix to Zermatt - It's probably the most famous ski tour in the Alps/Europe/the world and not surprisingly it gets busy. You'll hear stories of following 50 people up the skin track each day, and then skiing a line of bumps to a very full hut. 

The good news is that with a bit of planning (and maybe some luck) you can still do the Haute Route without the crowds. I did one trip in late February when we made it all the way to Saas Fee in 7 days but to be fair the weather was challenging and the huts were all closed, so maybe no surprise it was quiet. Then I finished my ski season with another trip, also very quiet but this time with perfect weather and open huts. We did a slightly unusual route with a couple of big days but covered all the main classic sections, booked huts last minute (none were more than half full) and skied some excellent spring snow. 5 days of excellent touring (by the time we reached Zermatt  the awful crust below Velan hut was long forgotten!) A great way to end a long and fun ski season.