Is there a limit in the number of layers for neural network? I heard the neural network has a problem with vanishing gradient problems even though the ReLU activation function is used.

In ResNet(that has a connecting function for reducing the problem), there is limit of maximum layers of 120~190(I heard).

For the complete AI performance(or general AI with strong intelligence) I believe that the limit of the number of layers must be solved.

Is there is any possibility that we find a new activation function that does not limit the number of layers? (maybe we could use exhaustive search... checking the train performance in a neural network of 200~500 layers)
The maximum number of layers that a network can have depends on many factors, such as the model architecture, activation function, optimization method, and others. For example, as you alluded to, ResNets is huge in increasing the number of layers possible, even up to 1000 layers for CIFAR-10. Another example is that when ReLU functions became more popular, Kaiming initialization allowed deeper networks to be trained by keeping the variance of each layer's activations at around 1, as the other popular method of Xavier initialization is more suitable for sigmoid activation functions.

So in general, there's no known limit except to keep experimenting and coming up with new techniques, including but not limited to finding better activation functions.