Can we predict next sample using the existing samples? Suppose that I have 18 data points and I'm sampling 3 data points each time. Suppose that I have 60 samples (each has 3 data points). Can we predict the next sample (of 3 points) from the existing sample? or the most recent sample?

In general you cannot do that unless you know the distribution behind the sample. For example, if the sample obey Cauchy distribution, then central limit theorem, strong law of large numbers, etc all fail. And you cannot use sample mean and sample variance even though $60$ is not a small data size. 
So the tools remaining are non-parametric. You can use tools like kernel density estimates to estimate the pdf and the cdf, as well as empirical likelihood functions so you can compute MLE in some sense. But these tools are not so easy to use and knowing when is appropriate can be subtle. 
