So this is a two-part question, because the mechanics in question are in the process of being changed.
The first part, as it currently stands, is that there is no maximum to any given "Connected" quality, although there is a point at which the term of recognition stops changing -- for instance, Society's highest title is "Toasted", at level 300, and The Revolutionaries' is "A Ringleader", at level 150. There are very few actions that require such high levels of Connected. The Widow, in particular, has no actions that require above 30 -- she's one of the least-used factions, alongside the Duchess.
However, as of writing, the various factions in London are being converted into a different system -- Favours and Renown, which you may have seen when dealing with Criminals, the Docks, the Tomb-Colonists, or the Rubbery Men. Under this system there is in fact a maximum level of connections.
Unlike the current Connected system, which just goes up and up in relatively minuscule amounts, Favours are granted discretely, to a maximum of 7, and generally only from opportunity cards. Usage of Connected items (such as a Nodule of Pulsating Amber for Rubbery Men, or an Engraved Pewter Tankard for Docks) then allows you to convert Favours into Renown.
Raising Renown becomes progressively more difficult and expensive each time you do it, and all Renown raisers require you to have a certain amount of one of the base stats (Watchful, Shadowy, Dangerous or Persuasive) based on your current level of Renown, which has the effect of soft-capping Renown at around level 50 for each faction.
Renown is a reliable way to raise certain Quirks above level 10, and can also unlock new factional content. For example, at Renown: Criminals 20, a new opportunity card becomes available, which provides both another source of Favours: Criminals and a very good way to cash them in (4.4 Echoes' worth of Prisoner's Honey for one Favour). Renown: Rubbery Men unlocks progressively better pieces of Persuasive equipment across London, although the final one requires access to the Fate-locked area of Flute Street.
There is no ETA on when any given faction will get converted -- these changes have been in progress since late 2015. When a faction is converted, your Connected score is exchanged for an amount of Favours and Renown, which is normalized based on the range of Connected among the whole player base (so that those with the highest score will begin with very high Renown, around 40 or higher).