The higher the better? My vote goes to the Andes.
There are lots of peaks over 6000 m that can be climbed without the need to cross a glacier. Challenging? Well, not from a climbing perspective. The main challenging thing about them is that they are high. And of course there is the weather factor: you can make it more challenging by climbing while there is a gale force wind pummeling the mountain.
An easy 6000-er, with often with good weather during the dry season, is
Chachani (6057m). The easiest routes are a walk up (F), but usually require crampons. Higher still is
Parinacota (6342m). Also a walk up (F), but a decidedly longer one, and nicer too. If you don't go too late in the season, you'll need crampons to get up there. I climbed the normal route in September 2009 and didn't (but I did on the descent, where we took a shorter, steeper route down the Bolivian side - still not very steep, mind you, just steeper). The highest,
Aconcagua (6962m), may require crampons, depending on the route and its condition, and the weather is always something to be reckoned with. Haven't climbed it myself though, so I'm not speaking from personal experience.
Challenging Andean peaks over 6000 m that don't require crossing a glacier? Haven't climbed any. There are plenty of possibilities, but you have to look beyond the normal routes for that. To get an idea of what I mean, have a look at the west face of
Sajama (6542m).
Sajama West FaceThere are two normal routes for Sajama, summiting from the left and the right, respectively. It depends on the conditions which one is the easiest, but basically they are both fairly easy (PD). I tried the NW ridge (on the left) but didn't summit because of high winds. If you climb the rocks as high as possible, there isn't all that much glacier to cross anymore to get to the summit (though this image, being taken from below, is somewhat deceiving). I wouldn't hesitate to cross that unroped: crevasse risk isn't the problem, it's the penitentes.
The reason I mention Sajama is this west face. Going up somewhere in the middle looks challenging to me. The more challenging the better, that's what you asked for, so here it is. The steepest part doesn't have a glacier. Despite that, personally I wouldn't want to climb it without a rope. Hec, I don't want to climb it at all, rope or not. Way out of my comfort zone! But if you say the more challenging the better, I say this fits the bill. I don't know if it's been climbed before, so perhaps you can make a first ascent.
Want something challenging, but higher than Sajama? Well, how about the north face of
Huascaran Norte (6655m)?
Huascaran Norte north faceHard to believe, but this face has been climbed - solo! Let me know when you've climbed it, and I'll let you adopt the page for Huascaran Norte.