Is it possible to increase the resolution of an image without expensive software?

Yes.

Something like this: SmillaEnlarger

Will do it without just making a blurry mess too. There are other fractal enlargers out there which try to mimic missing detail - but this one is free.

As Mike says, you can only work with the information that already exists in the graphic file, but that doesn't mean you can't increase the resolution, merely that you will have to make some trade-offs to do it. sometimes you are given an image that doesn't have the resolution needed for the purpose at hand, but have no choice in the matter. In those cases you can use the software like the one linked above (which increases the pixel count and averages the original pixels on either side) to reduce the apparent amount of blurriness to some extent. Then use existing techniques that can be found elsewhere on the web to increase sharpness and make the image appear sharper than it actually is. The result will definitely be less effective than using an image of the proper resolution, but might bring the image you are stuck with at least into the realm of usability.

Mike, I was taught the same absolutes as you, but sometimes it pays to step back from the instinctive, knee-jerk reactions, and check a little deeper. This is where learning and growing as a craftsman occurs.

There's a nice online one here http://waifu2x.udp.jp/ which uses "Deep Convolutional Neural Networks". It's especially good for enlarging logos and line art.