The golden ratio is achieved by any sequences of numbers generated by an arbitrary initial seed of two nonzero natural numbers by recursively adding consecutive numbers together, so it is a limiting property of a quite general class of sequences.
The Feigenbaum constants Itin in chaos theory have a similar nature.
Not so simply, the following sequences have important applications in mathematical physics.
The Riemann zeta function evaluated at natural numbers has several physical interpretations. See the responses to the Geometric, physical, probabilistic interpretations of the Riemann Zeta function for zeta(n > 1).
Scattering amplitudes in certain quantum field theories are related to the combinatorics of the associahedra. See references for OEIS A133437. (The generating functions for Fuss-Catalan numbers and their compositional inverses are particular cases.)
The soliton wave solution to the KdV equation for shallow water waves, which also crops up in string theory, is related to the Eulerian numbers. See the MO-Q "Why is there a connection between enumerative geometry and nonlinear waves?"
The coefficients of the Lagurre and Hermite sequences of polynomials have numerous combinatorial interpretations and are related to probabilty wave functions in quantum mechanics connected to orbitals and the harmonic oscillator.
The combinatorics of phylogenetic trees, used in modelling bifurcations in evolution of biological structures, are related to the Ward numbers. (There are other sequences in the OEIS related to genomics and structures of chemicals.)
Compositional inverse pairs of functions or formal series are related to flow equations characterized by autonomous ODEs, and the relations among series reps of these pairs can be couched in terms of the Euler polynomials (with integer coefficients) of the associahedra, the refined Ward numbers A134685, the refined Eulerian numbers A145271, or the noncrossing partitions A134264.