What is the electric potential when the test charge is placed very very close to the source charge, ie when $r=0$? Is it infinity?
In this case, why don't you just input actual values and see what happens?
You get the electric field by taking negative potential gradient and as you can see, it varies inversely with square of distance. So if the distance changes from 1m to 10^-3 m then the force will be 10^6 times greater. You see where I am going with this? It becomes stronger and stronger and hence you have to do more and more work. Thus the potential will approach infinity.