Building on @Pvl's answer, you can include type safety on your returned value as well if you use overrides:
function dig<
  T,
  K1 extends keyof T
  >(obj: T, key1: K1): T[K1];
function dig<
  T,
  K1 extends keyof T,
  K2 extends keyof T[K1]
  >(obj: T, key1: K1, key2: K2): T[K1][K2];
function dig<
  T,
  K1 extends keyof T,
  K2 extends keyof T[K1],
  K3 extends keyof T[K1][K2]
  >(obj: T, key1: K1, key2: K2, key3: K3): T[K1][K2][K3];
function dig<
  T,
  K1 extends keyof T,
  K2 extends keyof T[K1],
  K3 extends keyof T[K1][K2],
  K4 extends keyof T[K1][K2][K3]
  >(obj: T, key1: K1, key2: K2, key3: K3, key4: K4): T[K1][K2][K3][K4];
function dig<
  T,
  K1 extends keyof T,
  K2 extends keyof T[K1],
  K3 extends keyof T[K1][K2],
  K4 extends keyof T[K1][K2][K3],
  K5 extends keyof T[K1][K2][K3][K4]
  >(obj: T, key1: K1, key2: K2, key3: K3, key4: K4, key5: K5): T[K1][K2][K3][K4][K5];
function dig<
  T,
  K1 extends keyof T,
  K2 extends keyof T[K1],
  K3 extends keyof T[K1][K2],
  K4 extends keyof T[K1][K2][K3],
  K5 extends keyof T[K1][K2][K3][K4]
  >(obj: T, key1: K1, key2?: K2, key3?: K3, key4?: K4, key5?: K5):
  T[K1] |
  T[K1][K2] |
  T[K1][K2][K3] |
  T[K1][K2][K3][K4] |
  T[K1][K2][K3][K4][K5] {
    let value: any = obj && obj[key1];
    if (key2) {
      value = value && value[key2];
    }
    if (key3) {
      value = value && value[key3];
    }
    if (key4) {
      value = value && value[key4];
    }
    if (key5) {
      value = value && value[key5];
    }
    return value;
}
Example on playground.