  We demonstrate that standard delay systems with a linear instantaneous and a
delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior,
and weak ergodicity breaking if the nonlinearity is chosen from a specific
class of functions. In the limit of large constant delay times, anomalous
behavior may not be observable due to exponentially large crossover times. A
periodic modulation of the delay causes a strong reduction of the effective
dimension of the chaotic phases, leads to hitherto unknown types of solutions,
and the occurrence of anomalous diffusion already at short times. The observed
anomalous behavior is caused by non-hyperbolic fixed points in function space.
