I was playing recently the following game in 6max (max buy-in: 50BB) against a player quite often limping in (fish):
Preflop
MP, CO and BTN folds
SB: Fish (85BB) limps
BB: Hero (50BB) checks with 7♥8♠
Flop (2BB)
6♥4♠4⋄
check, check
Turn (2BB)
5♥
check, check
River (2BB)
3♠
Fish goes all in
Showdown Spoiler
As there are no flush possibilities and I had top straight, I called it. The opponent shown 34o and won the pot
I would definitely bet the turn, maybe 2/3 of this small pot. With a max straight you are ahead of the range of your opponent. You want to put money in the pot to let Villain draw for a flush (if he holds two ♥) or to a full (if he has a 4), or to a lower straight, all with negative EV. The only dangerous hands at the moment are quads or a made fullhouse and perhaps two ♥ with the 4♥. Now Villain should just call, and at least you have position on him to make the final decision. When he shoves, you are beaten by quads and by every fullhouse (33, 44, 55, 66, 34, 45, 46). One can argue that he would have raised preflop with a little pair in a blind battle, except that you mentioned that he limps quite often. So little pairs might be in his range. If you think that way, a fold is probably in order. But of course Villain's hand is very polarized: it is either a bluff or a hand that beats you and your decision will depend on your knowledge of this player.
I think you played the flop and turn okay. You could have bet the turn as well, the SB probably wasn't going anywhere with trips. If you got all your money in on the turn you would be ahead and only losing to 46, but of course you didn't know for sure if they had anything. While this is a hand you could reasonably put your opponent on, they have checked to you, so either they are slow playing it, or don't have the boat. Depending on my read I would probably tend towards the latter and would have bet.
However having reached the river, the fish suddenly going all in would raise alarm bells in my head. There are only 2BB in the pot, and now the pot is raised 50BB more. To me that suggests either a miracle river card for some monster hand, or a stone cold bluff. Either way, with only 1BB already at risk this would be an insta-fold for me. (This was my read before reading the spoiler text.) If I had previously bet the turn it would be different, and I would have a decision to make. Yes, there's a chance they are bluffing me, and I do have a really good hand, but the risk/reward ratio is off, and unless I have a good read that they are trying to push me around I would just release and live to outplay them in future hands.
First of all, I think betting the turn is better than a check. If he has nothing to pay you off with, that's fine, it wouldn't matter either way; but your hand is vulnerable with this board and the fact that the opponent could have basically any two random cards.
I fold the river. It should be pretty clear that villain's range here is very polarized--either he has nothing or he has a full house or better. It's easy to be a little blinded to the relative strength of your hand (hey, a straight is good!), but would you be so ready to make a call if you only had a 2? What if you had just A6? Because those types of hands have the exact same chances of winning against such a polarized range.
The main reason for folding is that he is making this wild bet out of position on that particular river card. From villain's point of view, you are holding two completely random cards--I mean, you haven't given him any information to go on during the hand (well, you probably don't have pocket aces or something like that)--the river makes it possible for some random hands to have hit a straight. Even though he may be a fish, is he betting 25X the pot on a bluff knowing that it's perfectly possible for you to have hit a straight? If you had checked to him on this river, it would be a little bit of a different story.
Another factor that could be overlooked is that if you're thinking "Hey, this guy isn't very good at poker, he could be making a stupid play", consider this: even if that is the case, if you fold, his money stays at the table and he'll continue playing stupidly. Some of that "dead money" will presumably funnel to you, so you could consider this a little extra equity that you have for just being at the table--you don't need it all in one hand when the risk is that you're wrong and lose a whole stack.
By the way, what I think that people are thinking here in villain's position is that they don't care about a stupid little pot, they're willing to gamble on the fact that the river hit you and that their bet looks suspicious enough so that you make a "heroic" call and they win a real nice pot. It's actually not so bad of a strategy for someone with the right image. They can bet 2BB's and have you call 20% of the time (I'm making up numbers here) for an expectation of 0.4BB's from the small bet; or, they try the big overbet wherein they only need to be a called a very small percentage of the time to be more profitable. If they're beat, then they're planning on losing their whole stack either way. I don't know if the fish literally think it out this way, but for a regular player who's playing other regular players this strategy is counterproductive from a meta-game consideration.
Maybe I am the fish but I would have called
On the turn there is straight and flush draw
6♥4♠4⋄5♥
Based on hero checking the BB both of those draws are in hero's range
With a set of 4s villain should have bet the turn to protect the hand
At this point hero should have protected from a flush draw with 1/2 to full size pot bet.
I did not add this until I saw the answer from Pin. You should give the check to Pin.
At the river I am only afraid of 64 as any other hand should have bet the turn. There is only 0.5% chance the villain has 64. Any pair should have raised pre flop.
Villain could be on
I get there is full house on the board but in my mind there is only one - the 64 that should get to the river like that. And the nuts has no reason to go all in.
Yes only 2 BB have been invested at this time but getting 2:1 on my money and I am giving 50% I am good. In a tournament that is fold but in micro I would take my chances and just re-buy. Table will not try and bluff me for a while if I am right.
Yes got beat by a miracle 3 on the river but in my mind that is disguised because villain played the hand poorly.
