What are the most challenging tasks aiming to achieve the lowest error rate? For example there is the MNIST database which is used to test artificial neural network (ANN), however it's not so challenging, because some hierarchical systems of convolutional neural networks manages to get an error rate of 0.23 percent.

Are there any similar, especially the most challenging tasks with dataset which are used as benchmark tests to challenge the AI which are fairly reliable and it's possible to pass, but most AAN are struggling to achieve the lower error rate?
Yes. Here are some of the most prominent ones and their respective state-of-the-art errors:


CIFAR-10: ~3.5% error
CIFAR-100: ~24% error  
STL-10: ~26% error
SVHN: ~1.7% error
ImageNet tasks: the best 2012 classification task solution got 15% top-5 error, better results are currently available


You can check an updated list of solutions here. Also, a more comprehensive list of modern datasets can be found here.