Expressions like "close to infinite" don't really have a mathematical or physical meaning. You can't get "close to infinite": any finite number is still an infinitely long way from infinity.
"Nearly the speed of light" actually falls into exactly the same category: no matter how fast you're going, you'll still need infinitely more energy to push you to the speed of light.
Still, the fact that they're both meaningless in a similar way is, well, meaningful. The closer an object (electron or otherwise) gets to the speed of light, the more mass it has. As the limit goes to the speed of light, the mass goes to infinity.