Lee and I both admire the T1 very much. He, however, is a bit more ecstatic about the sound than I am. An amp with perfect performance will only make up for the problems in the original source equipment. Sound can never exceed the sound quality that of the source. Ever. In the case of the DX100, the T1 cannot exceed the performance of the line out put or the headphone output. The T1 isn't a DAC. It is fed signal in an analogue signal. That signal goes into its circuits and is spit out to your earphones. To exceed the performance of a source like the DX100, or an iPhone, for instance, you have to use an outboard DAC, then feed that signal into an amp. (Of course, the DAC has to perform better than the source in putting 1's and 0's together, and has to then spit that signal out better than the source does. The DX100 has SPDIF output in optical and coax, which is a godsend if you really want to pull the last bit of performance from it.)
However, if the source has deficiencies, an amp, can overcome those deficiencies. There are many deficient sources out there. Notable examples are early iPods, current Sony Walkman players, Pre-2011 Cowon players, and many many more. The current iPhone 5 has a poor output compared to the iPhone 4. The DX100 has no apparent deficiencies however. The output is perfectly controlled with every earphone and headphone. Very little distortion, almost no background noise, perfectly linear frequency response. The T1's battery system may be better than the DX100's but if you are listening to earphones of any type, the DX100 has no issues. The T1's flavour comes in in a number of areas: distortion, stereo separation, background noise.
As much as I like the T1, the DX100 has better performance in a number of areas. But (please read this carefully), defects in audio are NOT always deleterious. A 100% perfect signal may sound flatter, duller, and less dynamic to your ears than one with defects. Consider valves (tubes): people swear by them. But they represent the audio niche with overall the worst measurable (close to the original recording) performance of any high end amp system.
The truth is that people want colour. While the T1 does NOT outperform the DX100 output, it does sound good. Mainly, it disappears, which is its job: the T1 isn't an amp made to really 'sound' good; it is very much an all-out performer meant to sustain the original signal as much as possible. Again, there is no way an amplified signal can be closer to the original at the source. It isn't possible with any amplifier. You amplify a certain signal.
If you photocopy a text, the best you can do with the best equipment is make it look as close to the original as possible. Same with amps. If the DX100 had major deficiencies running low Ω earphones, or lacked the current to supply even a semi-loud signal to headphone x without distortion, it would NEED an amp. As it is, there are very few headphones it can't technically 'handle'. Those headphones would likely stymie the T1.
That's not to say you won't enjoy the T1. It's just to say that the DX100 doesn't NEED an amp at all.