Do we have specific rules when we use four verbs in a sentence? It is grammatically appropriate to use four verbs in a row in a single sentence? Would this count as a four verb periphrasis?
Se ofreció irse comprando al retirarse.
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Propositions (or simple sentences) in Spanish have at most one finite verb (a verb that is conjugated) and can have non-finite forms like the infinitive, the gerund and the participle. I'd say it's impossible to place an upper limit to these because Spanish, like all languages, is recursive (well, most of them at least), and you can coordinate and nest non-finite clauses indefinitely. That is, you can say things like
"Ofreció dejarse observar caminando al salir sin demorarse pensando en volver."
In practice, you rarely see more than three or four verbs in a row, and these come from combining common patterns of verb + infinitive like dejar(se) + infinitive, perception verb (like ver, oír, sentir) + infinitive, and verb + gerund like quedar(se) + gerund. These phrases become difficult to parse when they get nested into each other, so we break them up. (Something like "Ofreció dejarse observar caminando" is about the longest I'd expect to produce or be able to parse.)
A verb periphrasis is a different thing. We call something periphrastic when it uses several free morphemes to express a single twist of meaning that could potentially be (or was historically) expressed by inflection. For example, the traditional Spanish passive voice consists of the verb ser plus the passive participle of the verb that we want to passivize (just as in English). Latin had a passive voice that was made by simply inflecting the verb. So grammarians call this Spanish passive voice pattern voz pasiva perifrástica, because it takes a complex verb phrase to express a single verbal feature. (The other passive voice we have is the pasiva refleja.)
Spanish also has a periphrastic future construction: ir a + infinitive (voy a comer, vamos a nadar). This contrasts with the simple future tense (comeré, nadaremos) which is shown by inflection. In this case both forms still exist in the language but they have different functions.
Addenda: I have just noticed that some teaching materials call all kinds of verbal constructions "periphrastic". These are patterns using two verbs (sometimes joined by a preposition) that have a special meaning. This is a slightly different use of the term. It's also rather broad, in the sense that it covers so many different types of combinations that it might not be useful except as a label.