What are some things that the government could offer for free?
1. WiFi
In the early 2000's I worked on several projects to create city-wide municipal WiFi networks, one was in Philadelphia, and it died on the vine, as did probably most of these projects. Still, the idea of ubiquitous internet access for free as an enabler for business and innovation seems sound. I have seen many communities with free WiFi in recreation centres, libraries and other municipal asset sites, but I'm thinking if it was everywhere, businesses could run point of sale and other services more cheaply, and Starbucks wouldn't be the only place you could get something done outside the home or office.
2. Water
You can get a drink of water from a public toilet or drinking fountain. I don't know a lot about homelessness, except that it is a multi-dimensional problem that is not easily solved. I'm pretty sure that the homeless have poor health outcomes, and hygiene can play a role in that (there is a campaign in Toronto to provide socks for the homeless, who apparently suffer from foot problems at an outsized rate). Imagine if there were publically available shower stalls (with curtains for privacy and a programmable locker to keep your clothes and belongings in) and the benefits it could provide the homeless (or really anyone who just needs a wash). Increasing dignity and health for the homeless could be a significant step in improving their situation.
3. Mental Health Service
Even within the pipe dream context of this list, the idea of a free hour of therapy for everyone in a given population is laughable. But as I mention in this Idea List part of the problem is that mental health services need to be more granular. Rather than an hour with a therapist, maybe what some individuals need could be served with 15 minutes with an intern. There are free hotlines for youth, for suicide prevention, but broadening the reach to the whole spectrum of mental health and utilizing every technological trick in the book (AI, video conferencing) is plausible, and would actually pay societal dividends.
4. Parks
Yes, there are green spaces that you don't have to pay for, but larger ones, some fairly small will still have a gate with an entrance fee. I usually pay these gladly, as I figure as long as people pay to use these facilities, they won't fall victim to budget cuts and be sold off and paved over. Still, if there's anything we should be encouraging people to do (if not outright paying them to do so) it's going outside for a walk (hike, bike, whatever). Not only is it healthy for them, but the time spent in nature is also time spent not polluting or consuming energy resources. Getting people reacquainted with nature is an important step in cultivating a culture where the environment is valued and protected. When it's costly to go hiking, people won't go.
5. Preventative Health Care
I'm Canadian. If I broke my leg right now, I could get it looked at and put in a cast for free (except for the cost of crutches, parking at the hospital or the ambulance). My mother is currently getting help from a personal support worker who helps her with washing and massage, provided by the government. The rationale is whatever services she receives at home are still cheaper than a full-time hospital bed. The system is imperfect and inefficient, but sound in principle. I'm middle-aged, and I have various aches and pains that slow me down and sometimes make it hard to sleep. If my issues had been screened and caught earlier, it would have been cheaper and less time-consuming to keep me running on all cylinders and thus a more productive worker and parent.
6. Universal Basic Income
Not an original idea, but too good to leave off this list. The failure rate of most new businesses makes it unattractive to most potential entrepreneurs if they need it to put food on the table and thus entrepreneurship tends to stay limited to a certain class of financially privileged people rather than anyone with a dream or idea.
7. Heat
Another threat to the homeless is cold exposure - this is especially true in Canadian cities. Subway grates are a common place to find the homeless sleeping. The heat from the subway system's ventilation system would be lost to the open air otherwise, it might as well do some good. Some cities seem intent on building architectural features to make spaces less hospitable for the homeless, but what about using technology to recapture lost heat from buildings and create warming spaces?

8. Swimming Lessons
I'm tired of hearing stories about adults who have drowned every summer.
9. An Apple A Day Keeps the Doctor Away
Imagine a vending machine that dispensed an apple. Scan some government ID or something so that you only get one a day (no hoarding), and make them ubiquitous and accessible. No excuses for not getting at least one helping of fruit.
10. Beer
...but only for me. The rest of you have to pay for your own.