High resolution magnetic resonance (MR) images are desired in many clinical applications, yet acquiring such data with an adequate signal-to-noise ratio requires a long time, making them costly and susceptible to motion artifacts. A common way to partly achieve this goal is to acquire MR images with good in-plane resolution and poor through-plane resolution (i.e., large slice thickness). For such 2D imaging protocols, aliasing is also introduced in the through-plane direction, and these high-frequency artifacts cannot be removed by conventional interpolation. Super-resolution (SR) algorithms which can reduce aliasing artifacts and improve spatial resolution have previously been reported. State-of-the-art SR methods are mostly learning-based and require external training data consisting of paired low resolution (LR) and high resolution (HR) MR images. However, due to scanner limitations, such training data are often unavailable. This paper presents an anti-aliasing (AA) and self super-resolution (SSR) algorithm that needs no external training data. It takes advantage of the fact that the in-plane slices of those MR images contain high frequency information. Our algorithm consists of three steps: (1) We build a self AA (SAA) deep network followed by (2) an SSR deep network, both of which can be applied along different orientations within the original images, and (3) recombine the multiple orientations output from Steps 1 and 2 using Fourier burst accumulation. We perform our SAA+SSR algorithm on a diverse collection of MR data without modification or preprocessing other than N4 inhomogeneity correction, and demonstrate significant improvement compared to competing SSR methods.