Welcome to the wonderful world of N-body gravity simulation! It's a field with a lot of potential.

Okay, this is a common problem when you're starting out. The force should approach infinity for a point particle - that is accurate. But even if it it's not a point particle, the force will get very large, and this will produce the same problem.

In "real life", as the free body approaches the fixed body, the force does get bigger and bigger. But then as the free body moves past the the fixed body, the force quickly changes direction while remaining big. You're now rapidly decelerating instead of rapidly accelerating, so in the end your final speed is the same as the speed you approached the fixed object with.

But in a numerical simulation, you can accidentally skip the second step, because you have a fixed time-step - you don't have infinite time resolution. So what happens is that you get a big jump in velocity, and that kicks you right out of the gravity field - you entirely jump over the step where you experience the gravity that's pulling you back down.

There are various ways to get around this issue. You can use a smaller time-step - this will reduce errors a lot. We also often use "softened" gravity. Here instead of F=GMm/r² we use F=GMm(r^2+e2) where e is a constant "smoothing length". This means that gravity doesn't go to infinity as r->0, but instead approaches some finite large number. Another thing is to force the energy of the free body to be constant. Calculate 1/2 mv² and -GMm/r and sum them to get the initial energy of the free body. If there are no energy sources, then this will be constant, and you can recalculate GMm/r each time-step to get what the magnitude of v should be. You can also (as others have suggested) just merge your particles if they get close enough, but that doesn't always work - often they still get close enough to make these numerical explosions even.