It doesn't seem natural to me to say 'People like keeping on working for other people'. Typically, we use: 'like + gerund' for actions we enjoy (like hobbies). For example: I like hiking. I like doing yoga. I like reading in my free time. While we use: 'like + infinitive' for actions we don't necessary enjoy but we think are important to do or necessary. For example: I like to wash the dishes before I go to bed (I don't enjoy this but do it out of a sense of duty). So I think it would be more natural to say: 'People like to keep on working for other people' (as this isn't something people necessarily enjoy, but maybe it is good for their careers). You can use 2 gerunds in a row. 'I recommend avoiding accepting strangers' sounds natural to me. Other examples could be: I recommend avoiding going to crowded places during the pandemic. I am considering setting up a business. I think typically 2 gerunds in a row sounds okay, but maybe 3 or more may sound a bit unnatural. (For example: Some governments are considering stopping recommending cloth masks).

For the first sentence I would say; 'People like to keep working for other people, or continue working for other people' The second sentence is fine, gerunds are simply verbs such as the word 'run' with 'ing' added to the end to change run to 'running', but yes, you can use gerunds more than once in a sentence, you could also say, 'I would avoid accepting invitations from strangers on social media', so you can use the 'ing' gerund form as long as it makes sense with what you're saying. Well done on your efforts to improve your English, you're a superstar. 😁 Also, you can use gerunds for something simple such as; 'I was walking in the park', but using 2 gerunds is also fine.