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As far as I know, after the verb "Start" we should use verb + ing structure. For example:
What if we use the verb "start" in present continuous?Is is correct to say/write :
Or we should change it in the following way:
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).
The three examples are grammatical, but the second is awkward and the others wordy. As StoneyB explained, horror aequi may account for people avoiding pairings as in “I am starting doing”. Here are further comments on your examples, intended to counteract the notion that “after the verb "Start" we should use verb + ing structure” by showing that starting by itself or a verb + ing structure by itself usually is a better choice.
• “I have to start doing the exercises” ‒ One might say this after a series of episodes of not doing the exercises. But in other cases, use only start or only do, as appropriate to circumstances. For example, if you've been putting them off, you might say “I have to start the exercises now”; or if you are telling a friend why you can't play at the moment, “I have to do the exercises”.
• “I am starting doing the exercises“ ‒ Say either “I am starting the exercises“ or “I am doing the exercises“.
• “I am starting to do the exercises” ‒ Again, one might say this after a series of episodes of not doing exercises. But usually either of “I’m starting the exercises“ or “I’m doing the exercises“ will better serve.
Your middle example is only marginally grammatical, if at all.
The reason is that the gerund makes the action progressive, and this is only needed once.
"starting to do" is a progressive action.
"start doing" is also progressive.
"starting doing" is a kind of "double progressive" which doesn't make sense.
However, it is not a general rule that gerunds do not combine. For instance
Starting running again after not doing it for several years can be difficult.
This is different because "running" basically serves as a noun here, referring to an activity. Contrast with:
My nose is starting {* running | to run }.
Here, running is not a noun referring to an exercise activity, but the a verb denoting moisture coming out of one's nose.
