OK.
I *THINK* I understand, now.
The mass is increasing dramatically . . . but only because of the dramatic amount of energy used to speed the protons up. And, still, though they are at a higher end of such speed-ups than ever before achieved, the percentages left between their highest and truly the speed of light are still significant distances to cover . . . and would require more or less infinite energy to get there—at least as we currently seem to understand things.
Is that close to right?
That's about it. Even though the distance between the speed achieved and "c" (the speed of light in a vacuum) is small as a matter of percentage of "c," the distance is infinite in terms of mass or energy (energy = mass times a factor).
Once relativistic speeds (say, 90% of the speed of light) are achieved, large energy inputs result in large mass increases, but small speed increases.
99.9999991% of the speed of light is not 99.9999991% of the way to "infinite mass." 99.9999991% of the speed of light in a vacuum results in a mass multiplier of about 7,500.