I'm probably not the greatest person to ask about amps given my relative inexperience.
As far as the ability to tailor the sound a bit more to my liking through different op amps and giving some extra volume headroom, the Lovely Cube does pretty well. But, depending on the op amps you use on the Titanium HD, you might get a
similar sound from it alone, although the Cube does help the sound fill out more compared to straight out of the Titanium HD. It has a few different gain settings that you can toggle depending on what your headphones need. I got the pre-assembled Premium and the construction is good, and the amp is also dead silent even if I crank the volume knob to the maximum (don't do this).
The stock op amp sounds too muddy to me. Switching it out for something like the OPA627BP helps tame the occasional sibilance on the DT880 while still giving enough detail. I have a few op amps for the Lovely Cube that I'll swap out to help different things. One helps smooth the high tones a bit, one helps boost the mids a bit and adds a little bit of forward depth, one widens the soundstage a bit more, one makes the bass a bit louder in relation to the midrange and treble, one adds a little bass boom (and removes some detail)... They're all slight variations on the same base theme, though. If you're interested in it, you should check out the thread (
http://www.head-fi.org/t/501046/the-lovely-cube-headphone-amp-lehmann-black-cube-linear-clone) for more and better impressions, and see if it sounds like something you're interested in. There's also a review at
http://www.stekkie.com/?p=156. The Matrix M-Stage is also another Black Cube Linear derivative. If you can find reviews for the Black Cube Linear (like
http://www.headfonia.com/lehmann-black-cube-linear/),
most of what applies to it will probably apply to the clones.
I think the general idea is that solid state amps are going to give you enough power to drive, will usually give a fast and precise sound but won't change the overall sound signature much, where tubes can add some warmth. But, that's just my uneducated generalization.