With respect to PaulMatthews, I disagree with his response #10. "John suggested a pet dog for Christmas" / "John kept a pet dog". In his second example, kept (keep) means to maintain; to provide support; to retain. In the OP, John does not 'retain' asking Bill. He continues the asking of Bill. It's not the same usage of 'keep'.
1. Suggest is a transitive verb. "Asking Bill" is its object- a thing that John does- a noun- a gerund.
2. Kept is intransitive in this usage. It's a helping verb, and asking is the participle.
Preliminary point: this discussion highlights one of the reasons why there is no point in retaining the traditional distinction between 'gerund' and 'present participle'. Endless disagreements abound about whether the
ing form of a particular verb in a particular clause is a gerund or a participle, resulting in learners being left either confused or misinformed. One alternative and widely accepted approach maintains that there is no viable distinction in function and calls the
ing form simply 'gerund-participle'.
My answer was intended to demonstrate that since a clausal complement of "suggest" and "keep" (like "asking Bill") can be replaced by a noun phrase, but not by an adjective phrase, "asking" must be a gerund and not a participle, at least in traditional grammar. In other words, the similarity between the two expressions is not that they have the same meaning (they don't), but that they head expressions with the same function (direct object). In other words "asking Bill" is closer to a noun that an adjective and hence is best analysed as a gerund, not a present participle.
The point here is that in traditional grammar, 'gerunds' are said to be like a nouns while 'present participles' are like adjectives. The latter can be clearly seen in an example like:
[1] People [earning $50,000 a year] don't qualify for the rebate."
[2] [Moderately affluent] people don't qualify for the rebate."
Here, the bracketed parts are alike in that they both modify the head noun "people". In [1] the brackets surround a clause with the verb "earning" as head; in [2] we have an adjective phrase with the adjective "affluent" as head. "Earning" and "affluent" are thus similar in that each heads an expression modifying a noun. But there is nothing in "John kept asking Bill" to indicate that "asking" is functioning like, or replaceable by, an adjective, and thus no reason to classify it as a present participle. (Incidentally, "kept" is not a 'helping verb' as you put it - not that that is relevant here).
Now for a better analysis:
"John suggested [asking Bill]."
"John kept [asking Bill]."
"Suggest" and "keep" are both catenative verbs and the non-finite clause "asking Bill" is functioning as catenative complement. In neither example do we attempt to claim that "asking" is a gerund or a participle - it doesn't matter - "asking" is simply a gerund participle verb-form heading the bracketed gerund-participial clause functioning as catenative complement of "suggested" and "kept".
For those not familiar with the term 'catenative', the word is derived from the Latin word for "chain", which is appropriate here since the construction consists of a chain of verbs in which all except the last have a non-finite complement.