Learn English – My favourite hobby is + verb + ing

The verb start takes complement clauses with either the marked infinitive or -ing forms, and there is no consistent difference in meaning between the two.

Consequently, both of your examples are grammatical. However, the version with the infinitive will usually be better, as it avoids using two -ing forms close to each other. This preference is generally acknowledged in contemporary linguistics and goes by the name horror aequi principle, a Latin phrase meaning, literally “aversion to the same thing”:

The horror aequi principle involves the widespread (and presumably universal) tendency to avoid the repetition of identical and adjacent grammatical elements and structures. —Gunter Rohdenburg, “Cognitive complexity and horror aequi as factors determining the use of interrogative clause linkers in English”, in Rohdenburg and B. Mondorf, Determinants of Grammatical Variation in English, 2003.

Another article in the same volume, U.Vosberg, “The Role of extractions and horror aequi in the evolution of -ing complements in Modern English”, states that

In many languages, we observe a strong aversion to the (immediate) co-occurrence of identical or similar grammatical structures. Thus, the establishment of -ing complements has been significantly delayed after matrix verbs appearing in the shape of -ing forms themselves. Such contexts tend to favour either the to-infinitive as verb complement or other alternatives (like nominal objects or finite clauses).

That's not the gerund.  That's its identical cousin, the participle.  If it were a gerund, then you would use the genitive form "my":

• The likelihood of my doing this is small

In this example, "doing this" is a gerund phrase.  "My" modifies the gerund phrase.  The modified gerund phrase is the object of the preposition "of".  
 

• The likelihood of me doing this is small

In this example, "doing this" is a participial phrase.  Participial phrases can modify nouns.  This participial phrase modifies "me".  The modified objective personal pronoun is the object of the preposition "of".  
 

Because there's no visible difference between the gerund and the so-called present participle, it almost looks like "me" and "my" are doing the same job.  They aren't.  The "me" acts as a modified object; the "my" acts as an object modifier.  Of course, in both cases you end up with some modified object for the preposition "for".  The overall meaning of each is much the same, and the choice between them is often no more than a question of stylistic convention.

If it helps, my personal preference is for the gerund construction.  I prefer the action, rather than the person, to be the literal object.