The font used for the Instagram logo is a custom modified version of Billabong. Here's the download link.

Billabong has its origins in the handlettered 40s and 50s script headings that seem to have endured especially in signage when style doesn't matter too much.

Unlike today's scripts Billabong is tightly spaced and Opentype font features allow for a myriad of ligatures to improve fit and evenness of color.

Users can select their choice of strokes, ornaments and ending flourishes for added emphasis and style.